Can a player gain priority and cast spells during the cleanup step?

The cleanup step is specifically designed to resolve without players receiving priority. Under CR 514.1 and 514.2, the active player discards down to maximum hand size and then all damage is removed from permanents and 'until end of turn' effects end — all without any player gaining priority.

However, CR 514.3 creates an important exception: if a state-based action would be performed, or if any triggered ability triggers during the cleanup step, those are dealt with before the step ends. In that case, players do receive priority and may cast spells and activate abilities as normal before a new cleanup step begins.

This can create a chain: if the new cleanup step also causes triggers or state-based actions, priority is again granted, and yet another cleanup step follows. This repeats until a cleanup step ends with no triggers and no state-based actions to address.

Concrete example: You discard a Squee, Goblin Nabob to hand size during cleanup. Squee's triggered ability triggers (it returns to your hand from your graveyard at the beginning of your upkeep — but if instead you had a card like a 'whenever a card is put into your graveyard' trigger), that trigger is placed on the stack, players receive priority, and spells can be cast before a fresh cleanup step occurs.

So the answer depends on whether anything triggers or a state-based action applies: normally no priority is granted, but if either occurs during cleanup, priority is temporarily opened to all players under CR 514.3.