Is it professionally viable to leave an American math PhD program for a Master's in Germany?
Is it likely that leaving my PhD program for a Master's in Germany after a year would damage my career prospects?
  


Yes: A Master's degree in Germany is considered by many people
 (and pretty much everyone over 50 or so) to be baseline education: In other words, it's the bare minimum you can have and expect to get a job in a relevant field. Yes, the times, they are a-changin', especially in e.g. software development, but "only" having a Bachelor's is still something of an anomaly: The Bologna Process has changed the structure of German university education but has not (yet) changed German culture and expectations.

Therefore, you'd be going from a culture with relatively lower educational expectations (the US, where a Master's degree hold a lot of weight and a PhD even more so) to a country with higher expectations and you're going down one educational level.

Likewise, it's possible that a well-known PhD program will be acknowledged in Germany in relevant field(s), but there are fewer German universities that are so amazingly well-known that they're acknowledged in the US. After all, the US is about the size of all of Europe, so it makes a bit of sense that the US is more insular than Germany is.


  
  In this or a similar set of circumstances, is it possible to leave one's PhD program without burning bridges completely?
  


It isn't exactly the same as leaving to get a Master's somewhere else, but it is possible that you could work on building a relationship and rapport with groups in Germany which do work similar to what you do and then ask them if you could spend some time there as a visiting researcher. This way you'd not burn bridges at your current department while still allowing you to experience how it is to live and do research in Germany. Also, you'd end up with a PhD in the end rather than just a Master's, which would make it much easier to move to Germany after the program should you wish to do so.