First, a test on the new Recharger Pistol animations. Supposedly they fixed the iron sights bug on it by making an entirely new set of animations just for it. However... it seems that the thirdperson animations are now bugged.
Here are the test instructions for the Recharger Pistol:
1: Obtain a Recharger Pistol, obviously.

2: Fire in 1stperson iron sights to affirm that you have the "fix". If it doesn't suddenly jerk off to the side after you fired, the bug is fixed.
3: Now, test fire in third person. Fire both from the hip and down sights. Look up and down to the extremes of your tilt elevations (IE: look up and down to the max) and fire. Does your character suddenly jerk back up/down to baseline elevation and look like they're firing straight ahead?
On to rifles... This works best if you have high Guns skill (optimally 100), are dosed on Steady, or both. You'll need a Hunting Rifle, Cowboy Repeater, or other similarly gripped rifle (IE: not a pistol grip rifle like the Service Rifle) in optimal condition (IE: above the Repair Threshold marker on its CND bar). Obviously, the Laser Rifle is out of the question. Precision-shooting rifles are preferred for this test (IE: not shotguns), and they must have iron sights, no scopes.
1: Look down sights in Third Person view. Be sure you're kneeling and have the above mentioned spread-reducing measures taken.
2: While still looking down sights, switch to 1st person view. You should still have the HUD crosshair revealed if all goes well. I am unsure if the quirk needed to test for this will reveal itself on the X-Box, but it does on the PC.
3: You'll have to look really closely to test for this: Is the HUD crosshair off of the iron sights' front notch post off-kilter? How far off-kilter? Vertical deviation isn't as important as horizontal.