Different scientific disciplines, including also the science of human movements - anthropokinesiology [Starosta, 2006], comprise issues that are vital and those that are marginal. 1. In various countries left-handed persons constitute on average from 5 to 50% of the entire population. As they have to live in a society of right-handers, they are forced to become adapted to them. Effects of such adaptation depend on individual predispositions of the person and on the attitude of the society to left-handed persons. 2. Depending on the progress of the adaptation process its consequences may vary. In many cases it was based on adaptation of left-handed persons to the surrounding society, i.e. becoming converted to right-handedness. As such conversion was conducted as a rule in an atmosphere of mental pressure and without knowledge of necessary teaching principles, it led to numerous disorders in functioning of the entire organism of a left-handed person. 3. The contemporary civilisation tends to give preference to right-handed persons. Left-handed individuals demand its modification which would take into account their functional distinction, and expect greater tolerance. 4. The present system of sport training had been prepared for right-handed persons. It has not been provided with a technique teaching programme or individualisation of their training which would be appropriate for left-handed persons. Consequently as left-handed persons have no other choice, they are forced to become adapted to the system. 5. There are various types of such adaptation. Sometimes left-handed persons achieve their biggest sport successes on the international arena with the help of the right hand (e.g. A. Grubba in table tennis or K. Date in lawn tennis) or thanks to being both-handed (e.g. M Strupler in handball). 6. An analysis of sport careers of various athletes, including also prominent ones, indicated the existence of great difficulties experienced by left-handed persons in adapting to training schemes designated for right-handed persons, and concurrently extraordinary adaptation abilities of the human organism. It also pointed to the existence of an unsolved problem of the distinct nature of sport training for left-handed persons, which, after all, concerns a significant population in the majority of countries worldwide. 7. A partial solution to this problem may be facilitated by application of an original concept for teaching and improvement of the sport technique based on movements symmetrisation (balancing the efficiency of both sides of the body) which allows maintaining the dominating side.