  We show that the Lorentzian Snyder models, together with their Galilei and
Carroll limiting cases, can be rigorously constructed through the projective
geometry description of Lorentzian, Galilean and Carrollian spaces with
nonvanishing constant curvature. The projective coordinates of such curved
spaces take the role of momenta, while translation generators over the same
spaces are identified with noncommutative spacetime coordinates. In this way,
one obtains a deformed phase space algebra, which fully characterizes the
Snyder model and is invariant under boosts and rotations of the relevant
kinematical symmetries. While the momentum space of the Lorentzian Snyder
models is given by certain projective coordinates on (Anti-)de Sitter spaces,
we discover that the momentum space of the Galilean (Carrollian) Snyder models
is given by certain projective coordinates on curved Carroll (Newton--Hooke)
spaces. This exchange between the Galilei and Carroll limits emerging in the
transition from the geometric picture to the phase space picture is traced back
to an interchange of the role of coordinates and translation operators. As a
physically relevant feature, we find that in Galilean Snyder spacetimes the
time coordinate does not commute with space coordinates, in contrast with
previous proposals for non-relativistic Snyder models, which assume that time
and space decouple in the non-relativistic limit $c\to \infty$. This remnant
mixing between space and time in the non-relativistic limit is a quite general
Planck-scale effect found in several quantum spacetime models.
