Between Tinny and stiknruda you have the answer.
The turn co-ordinator is a horrible, misbegotten mix-up of an instrument, which has the axis of its gyro tilted to sense rate of rall mixed with rate of yaw. A yank attempt to make a more quickly-responding instrument I believe (impatient nation).
In an erect spin yaw and roll are in the same direction. In an inverted spin from the point of view of the upside-down pilot and instruments the roll direction is opposite to the yaw.
Here's the catch : the turn co-ordinator sees both, so may respond to one, or the other, or neither. The pilot has senses that are fallible, and tends to see the roll more than the yaw. Since the yaw determines direction of spin and direction of required rudder to oppose the spin, this can be disasterous.
The turn needle will always give correct direction of spin. Hence good spin recovery training (for inadvertant spinning, not aerobatic intentioal spins) teaches you always to check your needle before recovery action, not to trust your senses.
I did a lot of early training on aerobatic types, which of course had turn and slip indicators, hence my dislike of turn co-ordinators.