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I suspect that in reality it was worn wherever the man wanted to wear it. A similar discussion can be had for artillerymen. Some would have put it on their chest, some over the shoulder, or on the hip, or on the horse, or wherever they liked as long as it was to hand in a hurry if needed.
Dear PDA
Interesting movie.
As I wrote officers have the respirator on their chests "infantry" style because they do not wear bandoliers. This is so in the film clip. It is by far the best place to wear it when secured properly. The troopers have it usually over one shoulder and from your film it looks like the right shoulder. If they are using the respirator there is only one place it can be;that is on or near the chest. Notice that the respirator bag is much higher than the haversack on my pic. If you don't know the equipment it is easy to confuse the two. The strap has to be the right length (quite short) for the mask to fit and for the hose not to be strained. Stowing it is another issue as you say it could be anywhere but it should be ready to hand if a gas attack was a possibility. Giving the choice I would wear it like the soldier in my pic over one shoulder- a simple solution when you have done you simply pull the bag so it rests on your back or under your arm. Trying it is believing.
BTW at the start of the film the horses look as if they have nose bags on - these are for feed. Called nose bags but they actually go over the mouth and the nostrils are not blocked.
The horses as depicted in the Strelets models and in reality did not have goggles so eye inflammation would have been a problem and the dutiful trooper would have covered his mount's eyes hence I say they would stay dismounted.