  A semantic adjustment to what physicists mean by the terms `special
relativity' and `general relativity' is suggested, which prompts a conceptual
shift to a more unified perspective on physics governed by the Poincar\'e group
and physics governed by the Galilei group. After exploring the limits of a
unified perspective available in the setting of 4-dimensional spacetime, a
particular central extension of the Poincar\'e group -- analogous to the
Bargmann group that is a central extension of the Galilei group -- is presented
that deepens a unified perspective on Poincar\'e and Galilei physics in a
5-dimensional spacetime setting. The immediate focus of this paper is classical
physics on affine 4-dimensional and 5-dimensional spacetimes (`special
relativity' as redefined here), including the electrodynamics that gave rise to
Poincar\'e physics in the first place; but the results here may suggest the
existence of a `Galilei general relativity' more extensive than generally
known, to be pursued in the sequel.
