VS2012 could open up VS2010 projects and leave them in a state where they would be usable in VS2010.
Does VS2013 have the same feature?
If you allow Visual Studio to upgrade the project automatically, you can open it in Visual Studio 2013, Visual Studio 2012, and Visual Studio 2010 SP1.
from following link
http://msdn.microsoft.com/en-us/library/vstudio/hh266747(v=vs.120).aspx
Visual Studio 2013 supports "round tripping" with support for solutions as far back as VS2010. There are some exceptions where an upgraded project may be required, depending on the type of project(s) in the solution. Generally, you can open and save solutions from VS2010, VS2012 and of course, any created in VS2013.
Based on the table from the MSDN link, it really depends on what your project contains. Some projects are able to stay cross-version compatible while others must undergo a conversion that limit or exclude the project's version compatibility.
For example, Coded UI Test projects can be opened across all platforms but a VS 2010 project containing a tie to MVC 2 will have to be upgraded to MVC 4 to work in VS 2013. That conversion makes it only compatible with VS 2012 and VS 2013.
I'm guessing there's a set of XSDs out there that can be used to test a project file's compatibility. I couldn't locate a site with the variant XSDs and haven't thoroughly checked through the specifications, but it looks like microsoft outlines them here: https://msdn.microsoft.com/en-us/library/vstudio/5dy88c2e(v=vs.100).aspx