People writing about the above question usually insist that the correct description of an about-face is as a 180° turn, and that those who call it a 360° turn are not thinking clearly: that a 360° turn would leave you facing the way you started. But in fact, there are two ways of measuring turns, and they give opposite answers to this question.
On the one hand, if you make a turn, and measure how much your new heading differs from your old heading, it is correct to say that a 180° turn constitutes an about-face, and a 360° turn leaves you facing the way you started.
On the other hand, suppose you are driving, and come to a roundabout (a little circle in the road that you have to drive through). If you go 180° around the roundabout and exit, then you are continuing in the direction you started. You must go 360° around to go back where you came from. Likewise, if you measure an angle between two lines, it is well-known that a 180° angle is a "straight angle", i.e., an unbent line; while a 360° angle, like a 0° angle, represents a reversal of direction.
What is behind the discrepancy between these two ways of measuring angles?
In the first approach, one is looking at the difference between one's direction coming out of the turn and the direction one had going into it. In the second, one is comparing the direction one faces as one comes out of the turn with the direction one faces as one looks back to where one came from. One could say that in each case, one looks from the present toward the future, but in one case one looks from the past toward the present, and in the other from the present toward the past. Both ways of viewing the past are reasonable, but they differ by 180°. The former way may seem more reasonable in describing motion; on the other hand, the latter fits with the traditional way of measuring angles in plane geometry.