Quarter-turning is a step in the production process of T-shirts or sweats that are knitted on a tubular cylinder. When the fabric is knitted and goes through the finishing process, it comes out as a flat tube of fabric. The flattening creates a crease on each side of the tube of fabric. Quarter-turning is when the tube is turned 1/4-way around before cutting the shirt's body. With this move, the crease is moved to the side of the shirt under the arm, rather than down the center front and back of the shirt. This results in a smoother surface for printing.
Source: Fruit of the Loom