Four tumblers on a rotating table

Four tumblers stand at the corners of a square table, each either upright or upside down, in an unknown arrangement. You are blindfolded. On each turn you may touch any two tumblers (two adjacent corners or two diagonally opposite corners), feel which way each is, and turn over either, both or neither. Then the table is spun to a random orientation, so you lose track of which corner is which. A bell rings the moment all four tumblers are the same way up.

Find a strategy that guarantees the bell rings within a fixed number of turns, whatever the starting arrangement and however the table spins. How many turns do you need?