  In this paper, we study the geometries given by commuting pairs of
generalized endomorphisms ${\cal A} \in \text{End}(T\oplus T^*)$ with the
property that their product defines a generalized metric. There are four types
of such commuting pairs: generalized K\"ahler (GK), generalized para-K\"ahler
(GpK), generalized chiral and generalized anti-K\"ahler geometries. We show
that GpK geometry is equivalent to a pair of para-Hermitian structures and we
derive the integrability conditions in terms of these. From the physics point
of view, this is the geometry of $2D$ $(2,2)$ twisted supersymmetric sigma
models. The generalized chiral structures are equivalent to a pair of tangent
bundle product structures that also appear in physics applications of $2D$
sigma models. We show that the case when the two product structures
anti-commute corresponds to Born geometry. Lastly, the generalized
anti-K\"ahler structures are equivalent to a pair of anti-Hermitian structures
(sometimes called Hermitian with Norden metric). The generalized chiral and
anti-K\"ahler geometries do not have isotropic eigenbundles and therefore do
not admit the usual description of integrability in terms of the Dorfman
bracket. We therefore use an alternative definition of integrability in terms
of the generalized Bismut connection of the corresponding metric, which for GK
and GpK commuting pairs recovers the usual integrability conditions and can
also be used to define the integrability of generalized chiral and
anti-K\"ahler structures. In addition, it allows for a weakening of the
integrability condition, which has various applications in physics.
