I don't have the Exchange 2010 media but I have the 2013 media. I need to downgrade, and as far as I can tell 2013 has downgrade rights. I have a 2k3 Exchange environment. I am going to install 2010, move resources, upgrade to 2013. Is getting the 2010 media the only way?
Yes, you'll need to get the media, but it's easily available:
You can use that to install Exchange 2010 (I included the base in case you really want just the base, but the sp3 media should install full version ok IIRC)
To downgrade, you'll need to reach out to MS Licensing or your MS partner to get a new product key to be used with Exchange 2010 as the one with 2013 you received won't work.
I'm pretty sure you need the actual media - It's not like the difference between versions of Exchange is all added files that you can just delete, there are substantial changes to the code base that make "downgrading" a 2013 system to 2010 simply impractical.
You might be able to get a download from Microsoft if you call them up.