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If a cleric casts Invisibility on his or herself, then uses their Channel Energy supernatural ability to cause damage to an undead creature, does the cleric become visible?

The way I've always played D&D 3.5 and subsequently Pathfinder, a new round starts at the after all of characters have acted in the previous round. If a character made an AoO without combat reflexes at any time in the round, they couldn't make another one until the next round started.

This week, one of the other players told me that wasn't quite correct. They said that in a given round, if a character took an AoO before their turn, they could take another AoO after their turn within the current round because subjectively for that character, the next round had begun.

Is this correct, and does any one know of a page in the Core Rulebook or SRD that clarifies this?