No, actually it doesn't. The ONLY orientation of the gyro that will cause it to be "backwards" is if you had it installed right side up, then flipped it over (or it was installed upside down, and you flipped it right side up). As long as the sensing axis is vertical and parallel to the main rotor shaft (that would be a line through the center of the gyro, going through the top and bottom of the case), you can mount the gyro any way you want. It can be rotated to any position around the vertical sensing axis and will work correctly.
Component orientation plays a part in your set-up
BUT -- not all servos move the same direction for the same rudder stick movement. JR servos move opposite direction from Futaba, for example.
If you switch out servos for another brand, you have a 50-50 chance that the new servo moves in the opposite direction than the one you replaced. Unless you have identical equipment in your helis, you simply can't set them up the same and expect things to operate in the right direction.
The Trex if built correctly requires that the TR servo PULL the TR pushrod forward to turn the nose to the right. Move the rudder stick, make sure the servo pulls the pushrod forward. If it doesn't, reverse the rudder channel at the transmitter reverse menu.
Pick up the heli. Turn its nose to the LEFT while watching the servo arm. The servo must pull the pushrod forward (commanding right turn when you move the nose to the left). If it DOESN'T pulle the pushrod forward, reverse the GYRO by flipping the NOR/REV switch on the gyro to the other position.
Of course, the heli has to be built right. Nose of heli to your right, tail to your left, tail rotor shaft is pointing at you. TR turns CounterClockwise when you turn the head in the correct direction. If it doesn't, your belt is twisted the wrong way. Fix it and make sure you only have a 1/4 turn in the twist.
The TR blade on the right side of the TR shaft is moving UP, leading edge UP, and the control ball is on the top. Fix it if it isn't working as described.
The TR blade on the left is moving DOWN, leading edge DOWN, and the control ball is on the bottom. Fix it if it isn't working as described.
And as described above, right rudder makes the servo PULL the pushrod forward (reverse the rudder channel at the TX reversing menu if it moves the wrong way). Then pick up the heli, turn the nose to the LEFT while watching the TR servo arm. It should pull the pushrod forward. If it doesn't, then change the REV/NOR switch position on the gyro body.
At this point the mechanics are correct and the transmitter AND gyro are set to run things correctly.
If you still have the uncontrollable spinning to the left, then either the front or rear TR drive pulley is slipping on its shaft. That would need to be fixed.
You say it spins as if it weren't in heading hold. That is nonsense, as well. A gyro when properly installed and setup, will NOT allow the heli to spin wildly. HH mode has nothing to do with it. The gyro can be set to operate in RATE (normal) mode and it will not allow the heli to spin. It appears that you don't really understand the meaning of "being in heading hold".
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