Can the active player cast spells during the cleanup step?

Normally, no player receives priority during the cleanup step, which means no spells or abilities can be cast or activated. The cleanup step exists solely to discard down to maximum hand size and remove damage and 'until end of turn' effects (CR 514.1, 514.2).

The one exception is if a triggered ability triggers during the cleanup step — for example, a 'at the beginning of the next cleanup step' trigger fires. When that happens, the triggered ability is put on the stack, players receive priority, and an entirely new cleanup step is performed after that one resolves (CR 514.3). During that window, players can cast spells or activate abilities as normal.

In practice, this exception is rare. Most cleanup steps simply end the turn with no opportunity for player interaction.

Example: The active player controls a permanent with an ability that reads 'At the beginning of the next cleanup step, draw a card.' When that trigger fires during cleanup, both players receive priority. The active player could respond to the triggered ability by casting a spell like Lightning Bolt before the trigger resolves. Afterward, another cleanup step occurs to finish ending the turn.