Oh fun, it has a name. Working on this right now with my mare, who gets stupid tight in her shoulders and stiff in her neck.
Here’s a discussion
http://myhorseforum.com/threads/thre…-doing.538212/
"1) Fléchi droit
This was essentially taking the historical flexions and applying them in movement, on the straight.
This is a classical movement that was included in both the French and German cavalry manuals.
German cavalry officer:

This movement is essentially, asking your horse to stay bent while moving on a straight line. It is a VERY good test to do on any horse and see if he is following his shoulders, or his nose. Obviously, in English disciplines, we don’t want the horse to follow his nose. The horse should be able to bend any direction and have his shoulders move regardless of this bend.
The easiest way to begin fléchi droit is by asking for a slight bend (with an open poll!!) in the corner or on a circle, and then moving straight while keeping this bend along the wall. As this becomes easier, start to ask for more bend. 45 degrees is the ideal, but in walk, you can go up to 90. [IMG2=JSON]{“data-align”:“none”,“data-size”:“full”,“src”:"http://myhorseforum.com/styles/default/xenforo/clear.png)
A rider showing about 90 degrees in walk:
[IMG2=JSON]{“data-align”:“none”,“data-size”:“full”,“src”:"https://scontent-a-atl.xx.fbcdn.net/hphotos-xpf1/t1.0-9/10443363_10202824436187801_7779792273862581291_n.jpg)
Personally, this is the very first movement I do with all horses immediately after mounting. For me, it is a ‘systems check’. It allows me to see right away if my horse is favoring a particular hindleg, contracted behind an ear, resistant to bend one direction, etc. When you start this movement, pay close attention to what your horse is doing. Is he twisting the poll? Trying to close his poll? Is he avoiding the bend by bringing in his haunches? OR, bringing in his shoulders? Is he offering too much bend? Is he resistant to bend? All of these attempts on his part are to avoid STRETCHING his outside muscles and it will help you to find out exactly where he is most contracted/where there is the most resistance.
Play with it and see what happens. Do it again after your normal schooling routine and see if it is any easier. If it isn’t- your ride did not address some areas of contraction in your horse. Even if it seems as though this is only addressing the neck- it is addressing the WHOLE body of the horse. It takes good balance and muscle suppleness for a horse to bend a clear 45+degrees and keep his four feet on the same two straight lines.
Remember that in this exercise your outside rein is critical in helping to keep the horse straight. Being able to open it towards the wall or close it towards the center of the arena (indirect rein) is crucial in controlling the shoulders. Also remember that ANY bend is useless of the horse is not holding it himself and just hanging on you. [IMG2=JSON]{“data-align”:“none”,“data-size”:“full”,“src”:"http://myhorseforum.com/styles/default/xenforo/smilies/no.gif) So allow your inside rein to be very soft and even slack occasionally to TEST that the horse is holding the bend on his own- sometimes, the horse will then show you a whole new evasion so you can see a way that he was trying to cheat the exercise. [IMG2=JSON]{“data-align”:“none”,“data-size”:“full”,“src”:“http://myhorseforum.com/styles/default/xenforo/clear.png)”