For built-in types, such as int, x++ yields the old value of x. There is no storage associated with this, so it would be impossible for the expression to be an lvalue.
In C, ++x yielded the new value of x (and was not an lvalue). Since the object x actually contains that same value, it is possible to have ++x designate the object x (i.e. be an lvalue). This is added functionality compared to yielding an rvalue, and the designer of C++ decided that this would be an improvement to the language.
For class types, it is possible overload any operator in such a way that use of the operator can yield an lvalue, xvalue, or prvalue. However it is considered good style to make overloaded operators have similar semantics to built-in operators, which is why it is normal for people to overload ++x to yield an lvalue and to overload x++ to yield an rvalue.
