  We consider a biconnection theory that extends general relativity, using the
recently defined mutual curvature as the fundamental object describing gravity.
To specify the two connections considered, we make a short detour into
information geometry, where a connection and its dual play a natural role. We
prove that the dual of a non-metric Schr\"odinger connection possesses torsion,
even if the Schr\"odinger connection itself does not. Furthermore, a manifold
with the dual of a Schr\"odinger connection is a quasi-statistical manifold.
With these two connections, we develop length-preserving biconnection gravity.
The field equations resemble those of general relativity, but replace the Ricci
tensor- and scalar with the mutual curvature tensor and mutual curvature
scalar, resulting in additional torsion-dependent terms. The covariant
divergence of the matter energy-momentum does not vanish in this theory. We
derive the equation of motion for massive particles, which shows the presence
of an extra force, depending on the torsion vector. The Newtonian limit of the
equations of motion is also considered. We explore the cosmological
implications by deriving the generalized Friedmann equations for the FLRW
geometry. They contain additional terms that can be interpreted as describing
an effective, geometric type dark energy. We examine two cosmological models:
one with conserved matter and one where dark energy and pressure follow a
linear equation of state. The predictions of both models are compared with
observational Hubble function values and the standard $\Lambda$CDM model.
Length-preserving biconnection gravity models fit well observational data and
align with $\Lambda$CDM at low redshifts $(z<3)$. They suggest that a modified
geometry could explain dark energy, late-time acceleration, and the formation
of supermassive black holes, as they predict different age of our Universe.
