Let's say I decide to keep a card with tentacles (madness) permanently at the end of the first round.
Bear in mind that Tides of Madness differs slightly from Tides of Time in one detail.
In Tides of Time, the card kept at the end of Round 1 is marked with a Relic token and is retained "for the remainder of the game." In particular, "at the end of the round [Round 2], players only take the cards played in Round 2 back into their hands while the Relic of the Past from Round 1 is left in their Kingdom."
In Tides of Madness, there are no Relic tokens. At the end of Round 2, you are allowed to take back into your hand the card you kept in play at the end of Round 1 (if you can even remember which one it was!).
Edit: this post was erroneous.