Recently I've started small research on the meaning behind Gagarin's Sturmanskie logo (as usual a mind blowing photo from ill-phil's priceless collection):
I know this is a rather simple (yet quite nice in my opinion) representation of Soviet Air Force navigator wings (insignia). I started my research from one of the military forums, where I found these scans presenting post WW2 insignia of Soviet Army (kudos for CABAL for posting it):
Soviet Armed Forces / Soviet Army (1946-1991)
As you can see there are two types of wings presented there, the one with the bomb and number 1 (1st class distinction) are Navigator insignia, similar to the one on Gagarin's watch and identical to "new" Sturmanskie logo. Gagarin graduated from 1st Orenburg Military Aviation College for Pilots in 1957 and was given the famous watch upon graduation.
But why Navigator wings since Gagarin was MiG-15 pilot before he endured his cosmonaut career? I believe (this is my opinion, I found the source but it's in Russian which I don't speak unfortunately) that it has something to do with first years of Orenburg College. In late 1928:
"The "Advanced Military School for Pilot-Observers" was transferred from Leningrad to Orenburg and was combined with the former Serpukhovsk school to make a joint Pilots and Observers (Navigators) school" 
Around 1941 the school was divided between Pilots and separate school for Navigators. But I believe that the insignia were kept unchanged in both colleges, probably to honor the tradition of original school.
The more information can be probably found in  but it's in Russian so I invite native speakers to maybe explain more details behind the Gagarin's military school and maybe watch itself.
 Orenburg Military Pilots School - OnAirpower.org
 ÷ïåîîáñ ìéôåòáôõòá --[ ÷ÏÅÎÎÁÑ ÉÓÔÏÒÉÑ ]-- ëÏÐÙÌÏ× é. ó. É ÄÒÕÇÉÅ. ïÒÅÎÂÕÒÇÓËÏÅ ÌÅÔÎÏÅ