  We undertake a detailed study of the $L^2$ discrepancy of rational and
irrational 2-dimensional lattices either with or without symmetrization. We
give a full characterization of lattices with optimal $L^2$ discrepancy in
terms of the continued fraction partial quotients, and compute the precise
asymptotics whenever the continued fraction expansion is explicitly known, such
as for quadratic irrationals or Euler's number $e$. In the metric theory, we
find the asymptotics of the $L^2$ discrepancy for almost every irrational, and
the limit distribution for randomly chosen rational and irrational lattices.
