$P 4_{2} 22$ tetragonal group choice of origin for symmetry of special projections
As stated in the International Tables of Crystallography vol A, as a rule the origin is locatedat a centre of symmetry or at a point of high site symmetry. Anyway, the choice of such a coordinate system is not mandatory since in principle a crystal structure can be referred to any coordinate system. In particular, all centrosymmetric space groups are described with an inversion centre as origin, whereas for noncentrosymmetric space groups, the origin is at a point of highest site symmetry.
In the space group n°93 the origin is at the intersection of 3 different 2-fold rotation axes (site 2a: 0,0,0), not at 0,0,1/4. I think this you're confused on this point, since the origin is always at 0,0,0 by definition.
From the trasnformation that you have assumed (y,x,-z+1/2), I guess you are referring to the operation number (7), that is a 2-fold rotation along the crystallographic [x,x,1/4] axis (not a plane! rotations are along an axis). As stated above, you can choose another coordinate system with a new origin shifted in respect the standard one, but any case the new origin will have again 0,0,0 coordinates in the new coordinate system. If you shift the origin, the coordinates of all the symmetry operations will be obviously redefined in the new coordinate system, since they are not independently distributed in space.