  In a multi-valley electronic band structure, lifting of the valley degeneracy
is associated with rotational symmetry breaking in the electronic fluid, and
may emerge through spontaneous symmetry breaking order, or through a large
response to a small external perturbation such as strain. In this work we use
scanning tunneling microscopy to investigate an unexpected rotational symmetry
breaking in Landau levels formed in the unusual floating surface band of ZrSiS.
We visualize a ubiquitous splitting of Landau levels into valley-polarized
sub-levels. We demonstrate methods to measure valley-selective Landau level
spectroscopy, to infer unknown Landau level indices, and to precisely measure
each valley's Berry phase in a way that is agnostic to the band structure and
topology of the system. These techniques allow us to obtain each valley's
dispersion curve and infer a rigid valley-dependent contribution to the band
energies. Ruling out spontaneous symmetry breaking by establishing the
sample-dependence of this valley splitting, we explain the effect in terms of
residual strain. A quantitative estimate indicates that uniaxial strain can be
measured to a precision of $ \lt 0.025 \% $. The extreme valley-polarization of
the Landau levels results from as little as $ \sim 0.1 \% $ strain, and this
suggests avenues for manipulation using deliberate strain engineering.
