24 pages. A tribute to Yujiro Kawamata, to appear in his sixtieth birthday conference proceedings, ASPM 2015. The webpage at http://homepages.warwick.ac.uk/staff/G.Brown/diptych.html contains links to auxiliary material. (First update adds more detail to calculations of polars and includes a new key variety to handle small cases all together. Only minor editing revisions in later update.)

Part I introduced diptych varieties $V_{ABLM}$ and gave a rigorous construction of them in the case $d,e\ge 2$ and $de>4$. Here we prove the existence of $V_{ABLM}$ in all the cases with $de\le4$. At the same time we construct some classes of interesting quasihomogeneous spaces for groups such as $\GL(2)\times\G_m^r$ based on the algebra of polars.