We prove that quadratical quasigroups form a variety Q of right and left simple groupoids and that the spectrum of Q is contained in the set of integers equal to 1 plus a multiple of 4. Properties of quadratical quasigroups are described and their inter-relationships are explored. Every element of a quadratical quasigroup is proved to belong to a 4-cycle. These results are applied to _nd conditions under which the group of additive integers, modulo n, induces quadratical quasigroups.