  We report high-resolution measurements of the in-plane thermal expansion
anisotropy in the vicinity of the electronic nematic phase in Sr$_3$Ru$_2$O$_7$
down to very low temperatures and in varying magnetic field orientation. For
fields applied along the c-direction, a clear second-order phase transition is
found at the nematic phase, with critical behavior compatible with the
two-dimensional Ising universality class (although this is not fully
conclusive). Measurements in a slightly tilted magnetic field reveal a broken
four-fold in-plane rotational symmetry, not only within the nematic phase, but
extending towards slightly larger fields. We also analyze the universal scaling
behavior expected for a metamagnetic quantum critical point, which is realized
outside the nematic region. The contours of the magnetostriction suggest a
relation between quantum criticality and the nematic phase.
