Straightening bent cylinder pins is a very tricky issue and the first
question that needs be asked is how many bent pins are there in the
cylinder?  How were they bent?

Most cylinder pins are bent by having a "run", which is usually
catastrophic to the cylinder.  It would also make the number of bent
pins so large that it's really not possible to begin to straighten them
out -- in which case it's usually needed to have the cylinder repinned
by a professional.  This is not cheap, and costs by the size of the
cylinder proportionally.

Assuming that there are only a few bent pins that need to be corrected
for whatever reason, I agree that you should first thoroughly
understand what is mentioned about this in Mr. Ord-Hume's excellent
works dealing with restoring musical boxes.

Any time that you attempt any repairs on any musical box, please
remember the first step is always letting down the spring.  This means
to remove all energy from the coiled spring, preferably in the accepted
way of slowly unwinding it, releasing the pawl, unwind, release the
pawl, unwind, etc.

Note: before letting down the spring, don't forget to remove the Geneva
stop.  You can think it's let down, but in all actuality you've only
reached the end of the Geneva stop!  If it is possible to let the box
play first, then by all means, let it play down to the minimum before
letting down the spring, but make sure that the spring has no tension
before proceeding.

Nancy Fratti,  http://www.nancyfrattimusicboxes.com/  of Canastota,
New York, sells (sold?) a small tool which is perfect for straightening
cylinder pins.  I found it to be perfect for this and many, many other
applications, so please contact her if you want this tool.  Otherwise,
you can find a syringe of the right internal diameter to sit nicely on
the pins, and cut it off cleanly so that you are only slightly longer
than the pins themselves (say 2 millimeters), and use this to support
the pins as you straighten them.

Good Luck!
Eli Shahar