Alternatively, just count three consecutive clicks that are close in time as one.
One slower but more UX-friendly option would be to allow long-press on a tile, it would zoom in, and you can pick left or right with each button.
Double tap is also an option but you need to remove the ability to select the content of the page.
That is, rotating with one finger by holding and dragging either in circle or straight line.
Better not the two finger rotation known from map apps.
Consistent across platforms, consistent "ideal" count, clever-ish game name
Cons: A little slower than simply clicking, maybe, and makes it easier to overshoot. Need to count rotations instead of clicks.
Pros: Quite intuitive (that's how dials work in most DAWs for instance) and would work well on mobile, even if the tile size is small.
A better option would be Ctrl-left-click for the other direction.