  Ultra-clean crystals of Sr$_3$Ru$_2$O$_7$ undergo a metamagnetic transition
at low temperatures. This transition shows a strong anisotropy in the applied
field direction with the critical field $H_c$ ranging from $\sim 5.1$T for
$H\perp c$ to $\sim 8$T for $H\parallel c$. In addition, studies on ultra-pure
samples revealed a bifurcation of the metamagnetic line for fields in
$c$-direction and it is argued that a nematic phase emerges between the
magnetization jumps. The aim of this study is to explain the field-direction
anisotropy of these phenomena. Based on a microscopic tight-binding model, we
introduce the metamagnetic transition by means of a van Hove singularity
scenario. We show that the rotation of the O-octahedra around the c-axis
expected for this material introduces a staggered spin-orbit coupling within
the planes and naturally leads to an anisotropy of the magnetic response. We
describe the low-temperature phase as a nematic state favored by forward
scattering processes. The spin-orbit coupling shows an influence on both, the
critical field $H_c$ and the occurrence of the nematic phase.
