The turn indicator uses a gyro with the X axis (the "axle"of the gyroscope) aligned with the lateral axis of the aircraft. It is held in a gimbal with its axis on the longditudinal aircraft axis. It can only sense yaw which through precession will roll the gimbal left or right from the pilots viewpoint. A simple mechanism drives a pointer on the instrument face. A spring stops the gimbal rotating and imparts a rate of yaw indication.
The turn co-ordinator has a similar mechamism, with the gyro spinning on the same axis, but the gimbal that supports it is tilted about 30 degrees up, so that its axis no longer coincides with the longditudinal aircraft axis.
If you think hard its possible to see that this arrangement makes the instrument sensitive to roll as well as yaw.
As for spinning.
In an upright spin(say to the left), when the flightpath can no longer be maintained and the nose starts to drop, full left rudder followed by full aft stick yaws the nose left, then the aircraft rolls left as it drops into the spin, during which it is continually yawing and rolling left.
The turn indicator will sense the left yaw and show a left turn. The turn co-ordinator will sense left roll and yaw and show a left turn. No problem there.
In an inverted spin, entry is the same, maintaining the aircraft flightpath with progressive forward stick until the stall break. In a spin to the right (from the pilots perspective) full right rudder will yaw the nose right, followed by a LEFT roll as the aircraft goes into the spin. In an inverted spin yaw and roll are opposite, relative to the pilot.
The turn indicator,which only senses yaw,will show a right turn. So rudder opposite to the needle as part of the recovery is the correct technique, upright or inverted.
The turn co-ordinator is unreliable . I can only tell you what it does in my aeroplane. In this case full right rudder will initially show a turn to the right, but then as the left roll developes(the wing on the pilots right drops towards the ground) the indication will change to a turn to the left,where it remains until recovery with full left rudder and back stick.
I would say that the old turn needle will get you out of the poo, but the turn co-ordinator might not.