  In the context of Lorentz-Finsler spacetime theories the relativity principle
holds at a spacetime point if the indicatrix (observer space) is homogeneous.
We point out that in four spacetime dimensions there are just three kinematical
models which respect an exact form of the relativity principle and for which
all observers agree on the spacetime volume. They have necessarily affine
sphere indicatrices. For them every observer which looks at a flash of light
emitted by a point would observe, respectively, an expanding (a) sphere, (b)
tetrahedron, or (c) cone, with barycenter at the point. The first model
corresponds to Lorentzian relativity, the second one has been studied by
several authors though the relationship with affine spheres passed unnoticed,
and the last one has not been previously recognized and it is studied here in
some detail. The symmetry groups are $O^+(3,1)$, $R^3$, $O^+(2,1) \times R$,
respectively. In the second part, devoted to the general relativistic theory,
we show that the field equations can be obtained by gauging the Finsler
Lagrangian symmetry while avoiding direct use of Finslerian curvatures. We
construct some notable affine sphere spacetimes which in the appropriate
velocity limit return the Schwarzschild, Kerr-Schild, Kerr-de Sitter,
Kerr-Newman, Taub, and FLRW spacetimes, respectively.
