  We discuss theoretically rather rear example of spin-flop transition which is
accompanied with changing the type of magnetic ordering and which seemingly has
not been addressed yet. We demonstrate that changing the type of magnetic
ordering can manifest itself in antiferromagnetic (AF) resonance experiments as
an apparent peculiar switching of the anisotropy at the transition from the
easy-axis type to the easy-plane one. We argue that this kind of spin-flop
transition is observed recently by K.Yu. Povarov et al., Phys. Rev. B 87,
214402 (2013) in Cu(pz)$_2$(ClO$_4$)$_2$ (where pz denotes pyrazine), one of
the best realizations of spin-1/2 Heisenberg AFs on square lattice having a
very small anisotropy. We show that the magnetic ordering changes at the
spin-flop transition in this material in the direction perpendicular to AF
square planes. We examine the microscopic mechanism of such behavior in
Cu(pz)$_2$(ClO$_4$)$_2$ and find that dipolar forces and extremely small
exchange coupling between spins from neighboring planes are responsible for it.
