After reviewing Mawaru Penguindrum once again I’m starting to feel that I mostly understand it. One thing that helped, since the story is told in a very non-linear fashion, was to construct a timeline of the events, including the back-story.
I’m posting it below with the warning that this is intended only for people who have already seen the anime. I’m not trying to explain the story here. I’m just putting the events in order. Reading this without having seen the anime first will just be confusing and quite likely will impair your ability to enjoy this magnificent work.
There’s a certain amount of guesswork here. Only a few specific dates are given, but many flashbacks include hints like “10 years ago” or “when I was 11” which let us at least assign a year. We can also make some guesses based on a knowledge of the Japanese school calendar and the age at which typical Japanese students enter a particular grade.
(And there are still some frustrating inconsistencies. In particular the foliage often doesn’t match the time of year implied by the dates. However Tokyo doesn’t get much snow and it’s possible that the Natsume estate is located well to the south, so this may not be as far off as it seems to someone like me who lives in the northern United States.)
Anyway, here’s my best guess:
1965-1970: The birth of Sanetoshi Watase. His red eyes and pink hair indicate that he has a special ability to understand and manipulate the workings of Fate. He grows up lonely and alienated.
1985-1990: Kenzan Takakura and Sanetoshi join a group of environmental scientists on an expedition to the Antarctic . The experience has a profound effect on them. They conclude that their society is irredeemably corrupt; a “frozen world” that must be “cleansed with fire.”
1987: The birth of Keiju Tabuki, Yuri Tokikago and Momoka Oginome. Like Sanetoshi, Momoka has pink hair and red eyes.
April 1994 – March 1995: Keiju, Yuri and Momoka meet in the third grade. Momoka begins to keep her diary and experiment with the Transfer of Fate spell. We know that she uses it 3 times:
- She saves the white rabbit and suffers a cut on her finger.
- She saves Keiju. The back of her hand is burned.
- She saves Yuri. She suffers multiple burns and is hospitalized.
Momoka falls in love with Keiju and imagines a future romance with him, which she writes down in the diary as if it were actually happening. She seems to believe that whatever she writes is fated to come true.
March 20, 1995:  The birth of Shouma, Kanba, Himari and Ringo.
Penguin Force launches its “Survival Strategy,” involving numerous bombs placed on Tokyo subway trains. Momoka detects the plot and tries to use the Transfer of Fate spell to prevent it. Sanetoshi casts a counterspell which has the effect of combining their karma and destinies.
The plot is only partially foiled: the subway system suffers severe damage and hundreds of people are killed or injured, but Tokyo is not “cleansed with fire.” Sanetoshi and Momoka both die but both continue to exist in a place between the worlds in the form of a “ghost” or “curse,” embodied respectively in the black bunnies and the penguin hats.
1996: It has proved impossible to repair the damage to the original Marunouchi subway line (the main loop connecting Shinjuku, central Tokyo and Ikebukuro). To replace it construction of the Tokyo Sky Metro begins.Kenzan takes over as leader of Penguin Force, which is renamed “Kiga Group.” The curse-form of Sanetoshi continues to direct the organization from behind the scenes.
Pingroup goes its own way as an independent above-ground organization, influenced by the curse-form of Momoka.
1996-1997: The birth of Himari, Hibari, Hikari and Mario.
2001: Shouma, Kanba, Masako and Mario are brought by their parents to Kiga Group meetings. (Presumably they don’t really understand what is going on.)
Kanba shares the Kiga Apple with Shouma, telling him that it is “the Fruit of Fate.”
At a later meeting Shouma wanders away and meets Himari, whose mother has vanished, leaving her alone in an empty apartment. Shouma shares the Kiga Apple with her. The Takakuras adopt Himari.
Kanba, Masako and Mario’s father, who is apparently about to attempt some dangerous mission for the Kiga Group, relinquishes custody of Masako and Mario to their grandfather. Kanba insists on remaining with him and is probably disinherited by his grandfather as a result.
Kanba’s father is killed and Kanba is adopted by the Takakuras. (The fact that he doesn’t go to live with his grandfather indicates disinheritance, or at least some sort of permanent break with the Natsume family.)
The Tokyo Sky Metro begins operations.
2002: Himari develops a dangerous fever during a typhoon. Kanba and his father carry her through the storm to the hospital.
Probably this is about when Masako’s grandfather died, though there’s not enough information to assign an exact year.
2006: Tabuki graduates from high school. After a conversation with him Ringo decides that she will “become Momoka,” follow the predictions in Momoka’s diary, and marry him. It is about this time that the Oginome’s marriage irretrievably breaks down.
Spring 2008: Himari, Hibari and Hikari agree to form an idol group called Triple-H. Their plans to audition are derailed by the accident that injures Himari’s mother. Presumably it is at this time that Himari teaches Hibari and Hikari the words of the Spell to Transfer Fate, without knowing what it is.
2008: The role of the Takakura parents in the terrorist attack is exposed. They go into hiding. (Actually they die, either by suicide or killed by the organization, but they continue to operate in ghost/curse form.)
Shortly thereafter Himari is forced to leave her school, either because of the scandal or because this is when she first becomes sick.
2009: Hibari and Hikari have a successful audition and begin their career as Double-H.
2010: Double-H become the mascots of Tokyo Sky Metro.
October 15, 2011:  Beginning of the anime series. Kanba, Shouma and Himari visit the aquarium and acquire one of the penguin hats. Masoko and Mario also visit the aquarium and get the other hat. (We see a glimpse of Mario at the aquarium with the hat in Episode 9.)
Himari collapses at the aquarium. While comatose her spirit meets Sanetoshi in the “library annex”. Himari is revived by the penguin hat. First appearance of the Princess of the Crystal .
[Not shown but implied] Mario also collapses and is revived by the other penguin hat.
October 16-19: (Episode 2) Himari comes home. Pingroup sends Penguins #1, #2 and #3 to the Takakuras. The Princess of the Crystal orders Kanba and Shouma to find the “Penguindrum.” The boys begin stalking Ringo who is stalking Keiju.
[Not shown but implied] Pingroup sends Esmeralda to Masako. Sanetoshi contacts Masako by telephone and tells her to get the diary. Masako begins stalking Kanba.
October 20, 2011: (Episode 3, Curry Day) Himari and Ringo have a Fated Meeting and become close friends.
Late October-December 2011: (Roughly Episodes 6-13.) Ringo’s plans are foiled at every turn. Keiju and Yuri become engaged and move in together. Ringo attempts a desperate gambit. Yuri steals half the diary and Shouma is hospitalized. In the hospital Masako kidnaps Shouma, confronts Kanba and ends up stealing the other half of the diary. Ringo and Shouma quarrel; Ringo makes a final attempt to seduce Keiju.
Himari has a relapse and is hospitalized. Sanetoshi appears, pretending to be a doctor with a special medicine that can save Himari.
January 2012: There seems to be a significant gap here, not shown in any episodes, during which Keiju and Yuri get married and presumably take time off for a honeymoon trip. We know they are married by Episode 14. Keiju would have wanted to schedule the wedding during his school’s January break. Since Ringo is shown in her school uniform in every episode, we can be pretty sure that none of the episodes take place during the January break.
Himari is revived by the medicine but remains in the hospital. She starts knitting scarves for Hibari and Hikari.
February 2012: (Roughly Episodes 14-20) Himari finishes the scarves. Sanetoshi sends them to Hibari and Hikari.
Ringo tries to reconcile with Shouma but is rejected. Yuri abducts Ringo but she is rescued by Shouma (really by Masako.)
Mario is possessed by the spirit of his grandfather. Both he and Masako end up deathly ill, but are revived by Sanetoshi.
Keiju abducts Himari and almost kills her, but relents at the last minute.
Himari is temporarily released from the hospital. Masako confronts Himari, forcing her to remember the truth about her past.
Meanwhile Hibari and Hikari record a song containing the Transfer of Fate spell.
Early March 2012: (Roughly Episodes 21 and 22): Kanba makes a pact with Sanetoshi: if Sanetoshi will save Himari’s life, Kanba will take command of the Kiga Group and redo the terrorist attack properly. A journalist from a Pingroup-owned media company is snooping around. Kanba orders the Kiga Group to kill him. Kanba and Himari both leave home.
Masako confronts Kanba. They retreat to a Kiga Group hideout but are surrounded by the police. Kanba is injured; Masako suffers fatal injuries trying to delay the police so that he can escape. Himari collapses and is hospitalized, presumably for the last time.
Ringo meets Hibari and Hikari and gets a copy of their song. Yuri returns her half of the diary to Ringo.
To prove his power, Sanetoshi revives Masako. Kanba takes her half of the diary. Kanba lures Ringo into a trap; both halves of the diary are destroyed.
March 20, 2012:  Execution of the Survival Strategy. The Kiga Group plants its bombs once again.
Final confrontation between Shouma, Kanba, Himari and Ringo. Thanks to the power of the Penguindrum everyone gets more or less what they deserve.
 On this date in our timeline a religious cult called Aum Shinrikyo launched a terrorist attack on the Tokyo subway system, releasing sarin nerve gas on a number of trains, killing 13 people and injuring hundreds more. This is a traumatic memory in Japan. Even the indirect references in this anime were considered rather daring.
 Shown as “today’s date” on the electronic sign in the library. (Episode 9)
 The English phrase “Princess of the Crystal” is used on the official website to refer to the combination of Himari channeling Momoka. It is not used in the anime itself.
 Shown as “return date” on the electronic sign in the library. Curiously, although the sign correctly lists October 15 as “Saturday” it incorrectly gives March 20 as “Monday” instead of “Tuesday.” Either there is some mystical significance to leaving out the leap-year day, or somebody messed up, though I won’t venture to guess whether the error was by the anime production staff or the fictional library staff.