  In this work we characterise the H\"ormander classes
$\symbClassOn{m}{\rho}{\delta}{\group,\textnormal{H\"or}}$ on the open manifold
$\group = (-1,1)^n$. We show that by endowing the open manifold $\group =
(-1,1)^n$ with a group structure, the corresponding global Fourier analysis on
the group allows one to define a global notion of symbol on the phase space
$\group \times \R^n$. Then, the class of pseudo-differential operators
associated to the global H\"ormander classes
$\symbClassOn{m}{\rho}{\delta}{\group \times \R^n}$ recovers the H\"ormander
classes $\symbClassOn{m}{\rho}{\delta}{\group,\textnormal{loc}}$ defined by
local coordinate systems. The analytic and qualitative properties of the
classes $\symbClassOn{m}{\rho}{\delta}{\group \times \R^n}$ are presented in
terms of the corresponding global symbols. In particular, $L^p$-Fefferman type
estimates and Calder\'on-Vaillancourt theorems are analysed, as well as the
spectral properties of the operators.
