iCloud calendar doesn't send invitations to Gmail users...
To clarify on the above post title: iCloud calendar invites sometimes send calendar invites to Gmail users, but not always.
It appears that if the user's Gmail account is associated with an AppleID account, then that user will not receive an iCloud calendar event invitation email. I discovered this when I began looking into why my wife did not receive an invite at her Gmail account, which also happens to be her primary AppleID for iTunes/AppStore purchases.
I found that my own primary AppleID, also associated with a Gmail account, did not receive invites, either. However, a test Gmail account I have, that has never been associated with Apple/iTunes/iCloud/etc., receives invites just fine. Also, MS Exchange mailboxes receive the invites properly, as do Yahoo accounts.
Any ideas as to what the secret is here? I cannot believe Apple would decide to prevent invites being sent out to 3rd party addresses that happen to also be AppleIDs on their own system, as that'd be quite ludicrous.