  We study ideal lattices in $\mathbb{R}^2$ coming from real quadratic fields,
and give an explicit method for computing all well-rounded twists of any such
ideal lattice. We apply this to ideal lattices coming from Markoff numbers to
construct infinite families of non-equivalent planar lattices with good
sphere-packing radius and good minimum product distance. We also provide a
complete classification of all real quadratic fields such that the orthogonal
lattice is the only well-rounded twist of the lattice corresponding to the ring
of integers.
