Symmetries are considered for a class of Hamiltonian models with one (spinfree) orbital per site. The models include common types of ParisierParrPople and valencebond Hamiltonians, defined over a continuous range of parametrizations. The symmetries investigated are linear canonical transformations and include the common pointgroup and alternancy symmetries. We find graphical symmetries extending the usual pointgroup symmetries and novel colorpairing symmetries which involve hybrids of pointgrouplike and alternancy symmetries of relevance for certain heteroatomic species. The occurence of recolorpairing transformations relating the eigensolutions of models for different molecules is also noted. Copyright 1990 John Wiley & Sons, Inc.