As far as I know (and I'd love to hear otherwise), one of the earliest examples of literary characters with incompatible arrows of time (as opposed to a simple reversed-chronology narrative) is from Lewis Carroll (no relation), in Through the Looking Glass. When Alice first meets the White Queen, she learns that the Queen experiences time backwards.
`I don't understand you,' said Alice. `It's dreadfully confusing!' `That's the effect of living backwards,' the Queen said kindly: `it always makes one a little giddy at first -- `Living backwards!' Alice repeated in great astonishment. `I never heard of such a thing!' ` -- but there's one great advantage in it, that one's memory works both ways.' `I'm sure MINE only works one way.' Alice remarked. `I can't remember things before they happen.' `It's a poor sort of memory that only works backwards,' the Queen remarked.
I agree, and I wish someone would do something about that. Carroll doesn't emphasize this device much in the book, but does offer one classic illustration of the phenomenon.
Alice was just beginning to say `There's a mistake somewhere-,' when the Queen began screaming so loud that she had to leave the sentence unfinished. `Oh, oh, oh!' shouted the Queen, shaking her hand about as if she wanted to shake it off. `My finger's bleeding! Oh, oh, oh, oh!' Her screams were so exactly like the whistle of a steam-engine, that Alice had to hold both her hands over her ears. `What IS the matter?' she said, as soon as there was a chance of making herself heard. `Have you pricked your finger?' `I haven't pricked it YET,' the Queen said, `but I soon shall - - oh, oh, oh!' `When do you expect to do it?' Alice asked, feeling very much inclined to laugh. `When I fasten my shawl again,' the poor Queen groaned out: `the brooch will come undone directly. Oh, oh!' As she said the words the brooch flew open, and the Queen clutched wildly at it, and tried to clasp it again. `Take care!' cried Alice. `You're holding it all crooked!' And she caught at the brooch; but it was too late: the pin had slipped, and the Queen had pricked her finger. `That accounts for the bleeding, you see,' she said to Alice with a smile. 'Now you understand the way things happen here.' `But why don't you scream now?' Alice asked, holding her hands ready to put over her ears again. `Why, I've done all the screaming already,' said the Queen. `What would be the good of having it all over again?'
Both Alice's Adventures in Wonderland and Through the Looking Glass rely on nonsense to tell a gripping story. Reversing an individual arrow of time is sufficiently nonsensical to qualify as automatically amusing, but also provocative. Why does everyone remember the same direction of time, anyway? (Actually that one's not hard to answer. If two systems with incompatible arrows were to noticeably interact, the one with more degrees of freedom would swamp the other one and quickly "correct" its arrow of time. No being that "remembered the future" would survive very long in the real world.)