  We introduce and study a new way to catagorize supersingular abelian
varieties defined over a finite field by classifying them as fully maximal,
mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers
of $A$ and its twists. We analyze these types for supersingular abelian
varieties and curves under conditions on the automorphism group. In particular,
we present a complete analysis of these properties for supersingular elliptic
curves and supersingular abelian surfaces in arbitrary characteristic, and for
a one-dimensional family of supersingular curves of genus $3$ in characteristic
$2$.
