Reciprocity is defined as: a mutual exchange of privileges; specifically : a recognition by one of two countries or institutions of the validity of licenses or privileges granted by the other.
If your DL is valid for towing RV doubles in your home state, it is also valid for towing RV doubles in ANY state that allows the practice of towing RV doubles, which is 39 states.....It is no different than one state saying you NEED a special license to drive an RV, but your state does NOT require that license. These are "driving privileges" nothing to do with equipment.
45 states are signers to the motor vehicle reciprocity compact and this is to protect your rights and not require you to hold 50 different DL's.
In CA, a resident can NOT tow RV doubles without a special license and endorsement, but because ID says my license is valid for RV doubles, CA does NOT say I have to have the same license as they require for their residents. I am protected by the reciprocity agreement.