  For a smooth manifold endowed with a (similarity) pseudo-Euclidean structure,
a stiff connection $\nabla$ is a symmetric affine connection such that
geodesics of $\nabla$ are straight lines of the pseudo-Euclidean structure
while the first-order infinitesimal holonomy at each point is an infinitesimal
isometry. In this paper, we give a complete classification of stiff connections
in a local chart, identify canonical models and start investigating the global
geometry of (similarity) pseudo-Euclidean manifolds endowed with a stiff
connection. In the conformal class of the pseudo-Euclidean metric g, a stiff
connection $\nabla$ defines a pseudo-Riemannian metric h such that
unparameterized geodesics of $\nabla$ coincide with unparameterized geodesics
of g but have a constant speed with respect to the so-called isochrone metric
h. In particular, we obtain a new natural connection on the open unit ball that
provides a compromise between properties of Cayley-Klein and Poincar\'e
hyperbolic models.
