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We determine the homotopy type of quotients of $S^n \times S^n$ by freeactions of $\mathbb Z_{/p} \times \mathbb Z_{/p}$ where $2p>n+3$. Much likefree $\mathbb Z_{/p}$ actions, they can be classified via the first$p$-localized $k$-invariant, but there are restrictions on the possibilities,and these restrictions are sufficient to determine every possibility in the$n=3$ case. We use this to complete the classification of free $\mathbb Z_{/p}\times \mathbb Z_{/p}$ actions on $S^3 \times S^3$, for $p>3$, by reducing theproblem to the simultaneous classification of pairs of binary quadratic forms.Although the restrictions are not sufficient to determine which $k$-invariantsare realizable in general, they can sometimes be used to rule out free actionsby groups that contain $\mathbb Z_{/p}\times\mathbb Z_{/p}$ as a normal Abeliansubgroup.

$\mathbb Z_{/p}\times \mathbb Z_{/p}$ actions on $S^n\times S^n$