  We give a classification of all principally polarized abelian surfaces that
admit an $(l,l)$-isogeny to themselves, and show how to compute all the abelian
surfaces that occur. We make the classification explicit in the simplest case
$l=2$. As part of our classification, we also show how to find all principally
polarized abelian surfaces with multiplication by a given imaginary quadratic
order.
