There is no parity on the Holey minx. If you consider switching centers via cube rotations then you can only switch by 72 degree rotations about the axis perpendicular to two parallel layers (U and D for example).
A 72 degree rotation of all pieces clockwise around the axis perpendicular to the U and D layers will perform four disjoint 5 cycles on corners, and six disjoint 5 cycles on edges. Overall this is an even permutation, and thus can be solved with an even number of quarter turns.
A 144 degree rotation of all pieces clockwise around the axis perpendicular to the U and D layers will perform four disjoint 5 cycles on corners, and six disjoint 5 cycles on edges - the same structure as the 72 degree rotation, but with different cycles. This is also an overall even permutation.
A 216 degree rotation clockwise is the same as a 144 degree counterclockwise rotation. A 288 degree rotation clockwise is the same as 72 degrees counterclockwise.
There are no other ways to do cube rotations other than rotating about an axis perpendicular to two parallel layers, so all ways of "cycling" centers will not create a parity error in either corners or edges.
Chris
