Quick A: No. But you can and often should overload func(X&) with func(X&&) (with const on the parameter if appropriate in either or both).

Here is the salient part of the SO question -- ignore the question's original title, because it's perfectly fine to overload pass-by-reference and pass-by-rvalue-reference where the former can be implemented using the normal copy-and-swap idiom:

Move assignment incompatible with standard copy and swap

[...] Here the assignment to a should use the "Move Assignment" operator. But there is a clash with the "Standard Assignment" operator (which is written as your standard copy and swap).

> g++ --version
Configured with: --prefix=/Applications/Xcode.app/Contents/Developer/usr --with-gxx-include-dir=/Applications/Xcode.app/Contents/Developer/Platforms/MacOSX.platform/Developer/SDKs/MacOSX10.9.sdk/usr/include/c++/4.2.1
Apple LLVM version 5.0 (clang-500.2.79) (based on LLVM 3.3svn)
Target: x86_64-apple-darwin13.0.0
Thread model: posix

> g++ -std=c++11 String.cpp
String.cpp:64:9: error: use of overloaded operator '=' is ambiguous (with operand types 'String' and 'String')
    a   = String("Test Move Assignment");
    ~   ^ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
String.cpp:32:17: note: candidate function
        String& operator=(String rhs)
                ^
String.cpp:54:17: note: candidate function
        String& operator=(String&& rhs)
                ^