|Your DVD drive only requires the power connector to be connected to it for it to be able to spin at least briefly when the computer is running and the button on the front of it is pressed to eject then insert the tray. It doesn't matter whether the IDE cable connector is backwards on a header on one header or what the jumper on the drive is set to in order to be able to do that.|
The IDE cable connector must be inserted into the drive and the mboard IDE headers in the right direction if it's possible to insert it either way, and the jumper setting must be correct for it's situtaion in order for the drive to work properly in Windows and to be recognized by the mboard's bios, but pressing the button on the front of the drive should still work even if that's wrong.
E.g. when they're on the same data cable, one IDE drive must be set to Master, the other to Slave or both must be set to Cable Select (if it's the IDE hard drive you're booting an operating system from it should be on the END connector, not the middle connector, when both drives are set to CS so that it's seen as master). Some IDE hard drives, e.g. Western Digital models, have two ways they can be set to Master - Master, single, or similar, for when they are by themself on a data cable, or Master, with Slave, or similar, for when they are on the same data cable as another IDE drive that's set to Slave - that must be set correctly for the situation, otherwise the Slave drive may not work in Windows or be recognized by the mboard's bios.
However, if the IDE data cable is damaged it could be two or more wires shorting together in the data cable are causing your problem.
It is common to un-intentionally damage IDE data cables, especially while removing them - the 80 wire ones are more likely to be damaged. What usually happens is the cable is ripped at either edge and the wires there are either damaged or severed, often right at a connector or under it's cable clamp there, where it's hard to see - if a wire is severed but it's ends are touching, the connection is intermittant, rather than being reliable.
Another common thing is for the data cable to be separated from the connector contacts a bit after you have removed a cable - there should be no gap between the data cable and the connector - if there is press the cable against the connector to eliminate the gap.
80 wire data cables are also easily damaged at either edge if the cable is sharply creased at a fold in the cable.
Try another data cable if in doubt.
If there's nothing wrong with the IDE cable, the only thing I can think of that can cause the DVD drive to NOT spin at least briefly when the button on the front of it is pressed to eject then insert the tray is you have it connected to the same data cable as an old proprietary CD-rom drive that has a 40 pin header but it is not wired the same as for an IDE optical drive.
I did that by mistake once and an IDE hard drive on the same data cable did not spin when the proprietary Panasonic 2X CD-rom drive was connected to the same data cable.
We HAVE seen a situation where some of the power connectors from the power supply have no +12v power, when the power supply has more than one +12v section - the DVD drive on a desktop computer requires both +12v and +5v power.