Minkowski space is a homogeneous affine real space and under this translationally invariant perspective that doesn't have privileged points it seems easier to consider its one-sheet 3-hyperboloid hypersurface description over the two-sheet one and this is something that just depends on a conventional choice of signature(timelike positive or negative). In this view it is hard to see that a natural or canonical choice of either past or future cone is possible just from the math of the Minkowski space independently of physical considerations that one might impose.
Even if by convention the usual two-sheet or even conical surface is used is there a canonical choice of H+ or H- sheet just from Minkowskian geometry that may be used to preferentially choose certain subspace inside one or the other sheet as the set of say the translations group action?