  Symmetries of a partial Latin square are determined by its autotopism group.
Analogously to the case of Latin squares, given an isotopism $\Theta$, the
cardinality of the set $\mathcal{PLS}_{\Theta}$ of partial Latin squares which
are invariant under $\Theta$ only depends on the conjugacy class of the latter,
or, equivalently, on its cycle structure. In the current paper, the cycle
structures of the set of autotopisms of partial Latin squares are characterized
and several related properties studied. It is also seen that the cycle
structure of $\Theta$ determines the possible sizes of the elements of
$\mathcal{PLS}_{\Theta}$ and the number of those partial Latin squares of this
set with a given size. Finally, it is generalized the traditional notion of
partial Latin square completable to a Latin square.
