That's an adapter to lap a valve by hand. See the two points on either end of that cross bar piece held on with the cotter pin? Those were fitted into indentations on the head of a valve. The female drive end was fitted to an L-handle or speeder, and the valve was turned back and forth with some kind of grinding compound poured inside.
Here are some photos of a T-11 with a different style adapter to engage the head of a valve.
I think you may have missed my reply to your original thread.
I'm pretty sure that what you have there is a partial Blackhawk Quick Disconnect (Q.D.) set from the late 1920's to early 1930's at the latest.
How are the sockets marked?
The early 1930's was a big transitional period for Blackhawk. They were phasing out the Q.D. technology and phasing in the "Lock-On" technology. At that time most of their sockets were 9XXX and 8XXX series, although they were still offering limited sets with the older Q.D. "T" model handles and DXX series sockets.
If you go back to Page 4 of this thread, I posted photos of a Q.D. 32CD set I found a few years ago. There are some other photos of other similar general service sets on that page, which all came in that gangster style attache case. CRTDI posted a catalog from 1934 that includes a mix of "Lock-On" and "Q.D." based socket sets. None of them have the T-15 crank, so you may have an older (early 20's) set, but I am pretty sure your kit should have at least 32 pieces. The smaller sets (19 and 22 pieces) came in a smaller, different case, with socket trays on either side. I have some other catalog scans from that era that I will check later for you.
I restored my set, to include a reproduction decal. Here's a thread on that project if you're interested, which also includes some catalog excerpts:
http://www.garagejournal.com/forum/showpost.php?p=2386150&postcount=1