For every positive integer N we determine the Enriques–Kodaira type of the Humbert surface of discriminant N2 which parametrises principally polarised abelian surfaces that are (N,N)-isogenous to a product of elliptic curves. A key step in the proof is to analyse the fixed point locus of a Fricke-like involution on the Hilbert modular surface of discriminant N2 which was studied by Hermann and by Kani and Schanz. To this end, we construct certain "diagonal" Hirzebruch–Zagier divisors which are fixed by this involution. In our analysis we obtain a genus formula for these divisors, which includes the case of modular curves associated to (any) extended Cartan subgroup of GL2(ℤ/Nℤ) and which may be of independent interest.