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A semantic adjustment to what physicists mean by the terms `specialrelativity' and `general relativity' is suggested, which prompts a conceptualshift to a more unified perspective on physics governed by the Poincar\'e groupand physics governed by the Galilei group. After exploring the limits of aunified perspective available in the setting of 4-dimensional spacetime, aparticular central extension of the Poincar\'e group -- analogous to theBargmann group that is a central extension of the Galilei group -- is presentedthat deepens a unified perspective on Poincar\'e and Galilei physics in a5-dimensional spacetime setting. The immediate focus of this paper is classicalphysics on affine 4-dimensional and 5-dimensional spacetimes (`specialrelativity' as redefined here), including the electrodynamics that gave rise toPoincar\'e physics in the first place; but the results here may suggest theexistence of a `Galilei general relativity' more extensive than generallyknown, to be pursued in the sequel.

A unified perspective on Poincar\'e and Galilei relativity: I. Special relativity