We show that if C is a supersingular genus-2 curve over an algebraically-closed field of characteristic 2, then there are infinitely many Richelot isogenies starting from C. This is in contrast to what happens with non-supersingular curves in characteristic 2, or to arbitrary curves in characteristic not 2: In these situations, there are at most fifteen Richelot isogenies starting from a given genus-2 curve.
More specifically, we show that if C1 and C2 are two arbitrary supersingular genus-2 curves over an algebraically-closed field of characteristic 2, then there are exactly sixty Richelot isogenies from C1 to C2, unless either C1 or C2 is isomorphic to the curve y2 + y = x5. In that case, there are either twelve or four Richelot isogenies from C1 to C2, depending on whether C1 is isomorphic to C2. (Here we count Richelot isogenies up to isomorphism.) We give explicit constructions that produce all of the Richelot isogenies between two supersingular curves.