I know its not a gruen but this case has similarities..the colored enamel notably the red/orange balls and of course the head. This serial # for this watch is 3994527 dates it to 1921. This is a swiss cased longines and the serial # is on both movement and case. Longines movement # dating is not perfect but is accurate enough to give you ballpark figure. My opinion is that the gruen is from the same time period as this watch.
The case maybe so, but I don't agree with dating the movement to 1921. The UV caliber was only introduced by Gruen in 1922 as Barney already said.
The only reliable way of dating Gruen pocket watches today is through Wadsworth case serial numbers which however were only used towards the end in combination with the UV calibre.
The earliest Wadsworth case serial numbers that I have found so far (total of 72 data points) are starting with “G0” which were produced from 1937 to 1939.
The lowest UV calibre serial in a “G0” case is 557,979 meaning that I can say with reasonable certainty that that movement was married to that case not earlier than 1937.
Bill’s serial number 557,314 is not too far off that serial number. However the reason I date it before 1934 instead of closer to 1937 is because Bill's movement still has “Adjusted to Temperatures” engraved on the bridges as explained before.
For reference the earliest UV calibre serial number I have found so far is 549,044 and the highest serial is 564,680 which is cased in a Wadsworth “G1” serial number (1940-1941).
This is my best educated guess of why I date it early 30's.