When using a class that has an enum property, one usually gets a naming conflict between the property name and the enum type. Example:
enum Day{ Monday, Tues
So long as the enumeration isn't nested within MyDateClass, I don't see that that's a problem. It's far from uncommon (in my experience) to have a property with the same name as the type it returns. I'll see if I can find some examples in the framework...
EDIT: First example: DateTimeOffset.DateTime (not an enum, but that's somewhat irrelevant)