graph isomorphism completeness of class X and not-X
Apparently, with graphs of class $X$ and not-$X$ you mean graphs that fulfill a property $X$ and those that don't. It is common that say a graph problem has a different complexity when the input graph is restricted to a certain class of graphs. In this sense, graph isomorphism is no different. As for your question, it is easy to come up with plenty of (even large) classes of graphs. For example, consider graphs that have an even number of vertices, and those that don't. Both are GI-complete, so at least in general, there are no implications.

If this is not satisfying, you need to address what you mean by "non-trivial graphs", but I don't think this would turn out to be too fruitful.