A quasi-platonic action of the group G on the Riemann surface S is a conformal action of G on S such that S/G is a sphere and the projection S->S/G is branched over three points. The surface S is called symmetric if there is an anti-conformal involution phi of S, called a symmetry. Equivalently, S has a defining equation with real coefficients. The fixed point subset or mirror of phi is a real curve. We are particularly interested in the case where G = PSL(2,q) and normalizes G-action. In this case S carries a tiling by hyperbolic triangles and the group generated by the reflection in the sides of the triangles is G^* = <phi> semi-direct G. In this talk we describe the quasi-platonic actions of PSL(2,q) admitting a normalizing symmetry phi. We address three questions about symmetries: The number of conjugacy classes of symmetries in G, the number of ovals in the mirror of a symmetry, and whether the mirror of the symmetry separates the surface.