  This paper introduces and investigates a class of noncommutative spacetimes
that I will call ``T-Minkowski,'' whose quantum Poincar\'e group of isometries
exhibits unique and physically motivated characteristics. Notably, the
coordinates on the Lorentz subgroup remain commutative, while the deformation
is confined to the translations (hence the T in the name), which act like an
integrable set of vector fields on the Lorentz group. This is similar to
Majid's bicrossproduct construction, although my approach allows the
description of spacetimes with commutators that include a constant matrix as
well as terms that are linear in the coordinates (the resulting structure is
that of a centrally-extended Lie algebra). Moreover, I require that one can
define a covariant braided tensor product representation of the quantum
Poincar\'e group, describing the algebra of N-points. This also implies that a
4-dimensional bicovariant differential calculus exists on the noncommutative
spacetime. The resulting models can all be described in terms of a numerical
triangular R-matrix through RTT relations (as well as RXX, RXY and RXdX
relations for the homogeneous spacetime, the braiding and the differential
calculus). The R-matrices I find are in one-to-one correspondence with the
triangular r-matrices on the Poincar\'e group without quadratic terms in the
Lorentz generators. These have been classified, up to automorphisms, by
Zakrzewski, and amount to 16 inequivalent models. This paper is the first of a
series, focusing on the identification of all the quantum Poincar\'e groups
that are allowed by my assumptions, as well as the associated quantum
homogeneous spacetimes, differential calculi and braiding constructions.
