A Humbert surface is a hypersurface of the moduli space
A
2
\mathcal A_2
of principally polarized abelian surfaces defined by an equation of the form
a
z
1
+
b
z
2
+
c
z
3
+
d
(
z
2
2
−
z
1
z
3
)
+
e
=
0
az_1+bz_2+cz_3+d(z_2^2-z_1z_3)+e=0
with integers
a
,
…
,
e
a,\ldots ,e
. We give geometric characterizations of such Humbert surfaces in terms of the presence of certain curves on the associated Kummer plane. Intriguingly this shows that a certain plane configuration of lines and curves already carries all information about principally polarized abelian surfaces admitting a symmetric endomorphism with given discriminant.