Could horse archers shoot to the right?
Horse archer has specific limits in comparison with foot archer. Bottom half of his body is fixed on the horse which cannot be easily turned, so if he needs to shoot somewhere, he usually has to turn upper half of the body there. If he is right-handed (holding bow in the left hand), he can cover roughly 180 degrees angle - from directly to the front, through agles at the left side of the rider, to directly to the back. By using jarmakee technique he can reach a bit further, but he always has a large angle on the right side thet could not be covered by just turning body.
Was horse rider really not able to shoot to the right?
I searched for ways to shoot to the right if you hold bow in the left hand. I expected that there will be some technique for it, but there is not. The only way how to shoot to the right (on horse) is to pass the bow to the right hand, and so shoot with the othe hand than you are used to. It is REALLY uncomfortable if you try it for the first time, but it can be learned. Even today there are horse archers that can do it (and use it in competitions). Yet most of contemporary archers cannot.
Could horse archers always shoot well with bow in any hand?
So let's look on historical horse archers who had really good motiuvation to be the best. How common was to shoot to both sides? Strangely there is very liuttle information about it. I could no fing answer on the internet, so I asked professional, namely Murat Özveri. And he told me some interesting facts about ambidexterity among Turkish archers.
First curiosity is that in Turkey there were so called Solaks, elite archers serving as sultan's bodyguard. Solaks formed 4 units containing 100 men each , and were chosen so that 2 units were created only from right-handed people, and 2 units from left-handed. It hints that left- or right-handedness was important for archer, and it was not expected that each archer will be similary good at shooting on both sides. If Turks really worked with left- or right- handedness in this way even in army organisation, then it really had some importance. Just for clarification - Solaks were mostly foot soldiers. But I expect thet if they worked with it at foot soldiers, then it had to be important at riders even more.
Second curiosity is that in historical texts there is sometimes literally said that archer should train shootinf even by the other hand than is batural for him. I did not find it in either Aeap archery or Saracen archery, but Murat Özveri said that there are more historical texts, but they were not translated to English. And there it is written.
Conclusion is thathorse riders (at least Turkish) were really better on one side than on the other (otherwise it would not make sense to take care about left- and right-handedness). But at tyhe same time thay had to train even the less handy side. Murat Özveri so expects that horse archers always tried to have enemy on their left side; but if it was not possible, they were able to shoot even to the right. It was used rather for emergency shots, weaker and to the shorter distance, but they ware able to endanger enemies even on the right side.