  We address the ground-state properties of the long-standing and much-studied
three-dimensional quantum spin liquid candidate, the $S=\frac 1 2$ pyrochlore
Heisenberg antiferromagnet. By using $SU(2)$ density-matrix renormalization
group (DMRG), we are able to access cluster sizes of up to 128 spins. Our most
striking finding is a robust spontaneous inversion symmetry breaking, reflected
in an energy density difference between the two sublattices of tetrahedra,
familiar as a starting point of earlier perturbative treatments. We also
determine the ground-state energy, $E_0/N_\text{sites} = -0.490(6) J$, by
combining extrapolations of DMRG with those of a numerical linked cluster
expansion. These findings suggest a scenario in which a finite-temperature spin
liquid regime gives way to a symmetry-broken state at low temperatures.
