This clang tidy check checks for classes that violate the rule of five and zero as specified in CppCoreGuidelines: https://github.com/isocpp/CppCoreGuidelines/blob/master/CppCoreGuidelines.md#c21-if-you-define-or-delete-any-default-operation-define-or-delete-them-all.
If a class defines or deletes a default operation then it should define or delete them all.
I don't have a better idea for a name, but this doesn't seem to tell the user much about what the check will do, either.
Would it make sense to expand this check into a rule of (2)/3/(4)/5 check (http://en.cppreference.com/w/cpp/language/rule_of_three) with config options as to which rule to enforce, so that it optionally includes the destructor? I'm thinking of config options like: RuleOf=3|5, IncludeDestructors=true|false with the defaults being 3 and true, respectively? The check could be named misc-cpp-special-member-rule (or something more clear than that). As it stands, the check currently handles the rule of 2 and 4, but I think a lot of people may want the rule of 3 or 5 instead and it would be good to cover them all under the same check name.
Note, this could be done in a follow-up patch, I am mostly interested in getting the name correct for the check.