  Strong interactions that favor chiral p-wave pairing, combined with strong
pair breaking by confining boundaries, are shown to lead to new equilibrium
states with different broken symmetries. Based on a strong-coupling
Ginzburg-Landau (GL) theory that accurately accounts for the thermodynamics and
phase diagram of the bulk phases of superfluid $^3$He, we predict new phases of
superfluid $^3$He for confined geometries that spontaneously break rotational
and translational symmetry in combination with parity and time-reversal
symmetry. One of the newly predicted phases exhibits a unique combination of
chiral and helical order that is energetically stable in cylindrical channels
of radius approaching the Cooper pair coherence length, e.g. $R\sim
100\,\mbox{nm}$. Precise numerical mimimization of the GL free energy yields a
broad region of stability of the helical phase as a function of pressure and
temperature, in addition to three translationally invariant phases with
distinct broken spin- and orbital rotation symmetries. The helical phase is
stable at both high and low pressures and favored by boundaries with strong
pair-breaking. We present calculations of transverse NMR frequency shifts as
functions of rf pulse tipping angle, magnetic field orientation, and
temperature as signatures of these broken symmetry phases.
