It is a common question and the answer is no, Revit does not save backward, to an older version.
People often regard saving a file from one format to another as a trivial matter. As software evolves and new features are introduced these new things have no equivalent representation in the older format. Even Microsoft Word or Excel warns us when we save a file to an older version and those are much simpler elements.
For example, when Revit Parts were introduced. If we could save to an older version that did not know what Parts are how should that transition be handled? What should the developers decide to do with them? What can they become and retain some usefulness?
With AutoCAD there are new features that can survive a trip backward, like Tables can be reduced to text and lines so it still looks like a table but is really less elegant than the Tables they came from. If the file is then saved in the most current version there is no "make me back into a table assumption" so fidelity is lost.
It is difficult to allow for forward saving (upgrading) too. Nobody is very pleased if they upgrade a file and something breaks. Saving backward is most assuredly going to break elements the more object oriented our design tools become and evolve.
Revit's founders chose to eliminate the complexity and development distraction of saving backward at the outset. Most software like Revit does too even if they don't acknowledge it outright. They may allow saving backward in concept but there is always some loss of fidelity or utility in the process.
Revit does permit exporting data to other formats to permit it being referenced in some way by other tools but it remains impractical to expect casual backward and forward file translation.
When Revit was introduced in 2000 it was a rental, we paid monthly and internet access was required. Using the latest version was expected, required. That's never changed even when Autodesk purchased Revit Technology Corporation in 2002. They just made it possible to buy a perpetual license like their other software. Conceptually though nothing has changed.
Autodesk has a legacy of customers who transition from AutoCAD where it is normal to ignore a new release for several years before upgrading. That is possible because the rate of change for AutoCAD has generally been much less aggressive than with Revit which is much younger and has different objectives.
It is easy to have access to the latest version at all times, via subscription. Fwiw Autodesk recently announced that new licenses will soon be all subscription based (rental) going forward so the concept that Revit embraced in 2000 has come full circle.
I'm not always pleased with Autodesk's choices and how they affect me as a customer but no company is perfect. For example, I'm writing this post on an iPad that can't just give me bloody arrow keys (without an external keyboard). Instead I have to use a goofy fussy magnifying glass to reposition the cursor.
