Teisco was a Japanese manufacturer of affordable (cheap) musical instruments from 1948 until 1969. They were imported into the US sometime around 1950 and were often rebranded by other "affordable" guitar companies. Teisco typically copied the popular guitar designs, but the T-60 was an original design that had a "monkey" grip in the body, and the cutout in the headstock.
Here is a page from the Teisco catalog showing the T-60:
I am not sure why Glen took a liking to this guitar. It certainly was inferior to the Fenders and Gibsons, but he showcased it in the early 60's on Star Route
as well as other shows.
Additionally, as witnessed by this picture
, Glen used the Teisco while with the Wrecking Crew, as well as HERE
Glen soon moved on to using a Fender electric, Mosrite, Ovation, back to Fender, then to G&L...but it is apparent that Glen had fond feelings for his T-60 because it was one of the guitars he choose to surround him in this picture that was taken on his staircase in Malibu: