While I'm glad you got to the right conclusion, what Weirdo and I have been trying to impress is that "there is no restriction against..." is a fundamentally incorrect interpretation of any possible rule in BattleTech. Or at the very least way of approaching rules questions.
And we have a fundamental disagreement on that. I'm not trying to make my mech use the Care Bear Stare on you ("the rules don't say I can't do it!"). I am simply reading the literal language of the rules.
Let's take a look at the original Battletech Manual: The Rules of Warfare (with the Atlas on the cover). I picked this one at random, but it doesn't really matter, because most versions of the rules are virtually copy/pasted from it. In the section "Playing The Game", Combat, page 17.
After completing the Movement Phase of the turn, the players begin to engage in combat. There are two forms of combat: Weapon Attacks and Physical Attacks. Weapon Attacks are attacks using the unit's armaments, such as missiles, lasers, and autocannons. In Physical Attacks, the 'Mechs use their weight to inflict damage on targets.
(skip description of how damage works)
During the Weapons Fire Phase, players use the armaments of their units to inflict damage on targets. For one unit to fire at another, it must have a clear line-of-sight to the target, and the target must be within the range and firing arc of the weapons that player wishes to use. The likelihood of a shot hitting a target is then calculated based on the range, movement of the target and firer, intervening terrain, and other factors.
Hmm, well, it appears to shoot at a target you have to have line of sight, be within range, and the target has to be within your firing arc. But it doesn't say anything about who or what you can target. According to that, "units" can fire at "units".
Maybe it says something in Line of Sight? I'm not going to quote the whole section, but it doesn't. What about the section on Firing Arcs? No, it doesn't say anything there either. What about in the "Firing Weapons" section on page 19?
FIRING WEAPONS
After a player has determined that a target is within the firing arc of his weapons and there is a clear line-of-sight to the target, his unit may fire.
The phrases "enemy unit" or "friendly unit" never appear in the book, not that I've found. The phrasing is either 'mechs shooting at 'mechs, units shooting at units, or attacker shooting at target. And that's the case with Battledroids, Battletech Third Edition, Battletech Fourth Edition, The Rules of Warfare, Battletech Master Rules, and Battletech Master Rules Revised. Even Total Warfare uses those same terms except for the definition of "target" that I found on page 42 (posted above).
The game allows a unit to shoot at another unit. It makes no distinction as to who controls either one. Saying "well obviously you can't shoot your own guys" is you reading something into the rules that (until Total Warfare page 42) was not there. The game also allows you to make facing changes, but it doesn't use the terminology "left" or "right". I don't think anyone would argue that you can't turn left.