Fold the tip of each end over by an inch or so and snip a vertical slit in the center that is about half an inch long. Unfold both tips and and lay them on top of each, holding the larger ball's strip on the left and bottom and the shorter strip on the right and top.
Butt them head to head as shown and then overlap them right side to right side, shorter strip on the right and top. Poke the tip of the right and top strip through the slit of the strip underneath. Once it is inserted by a couple of inches, loop it back on itself and insert a further back portion through its own slit.
Carefully pull the entire length of the sleeve strip through its slit.
Fiddle with the two ends as necessary, pulling both straight so they are not folded over on themselves but are nested together and extended in both directions.
Pull tight--the junction should practically disappear.
The neck section will yield some more irregular yarn and bumpier shoulder seams but I think that adds interesting texture. After removing the neckband, just cut around and around the neck opening until you run into the sleeves. When you're done, there shouldn't be much t-shirt left!
Now that's recycling!