No worries Gordon!
Yes, if your tunic is from a cavalryman, there is no need for a Reiterabzeichen in bronze since his riding training and qualifications exceeded a Reiterabzeichen III.
Soldiers and Officers of other arms, like Infantry, Signals, and Police etc. had to qualify for horseback duty through the Reiterabzeichen program of the Reichsverband für Zucht und Prüfung des Deutschen Warmbluts.
The Wehrmacht or SS put their candidates for mounted duties through a riding course given by a Riding Instructor. Then the candidates had to take a three-part exam consisting of Horsemanship-theory, Dressage riding and Jumping over fences. All three exams had to be passed to be awarded the Reiterabzeichen Klasse III (bronze).
By wearing the Reiterabzeichen a Soldier, NCO or Officer showed his mounted duty qualification.
The silver Reiterabzeichen Klasse II could be awarded by either taking an advanced-riding-skills examination or through merit (winnings in either Jumping, Racing, Eventing or Dressage competitions on a national level.
The golden Reiterabzeichen Klasse I was only awarded through winnings in Competitions of the highest levels.
The Reiterabzeichen qualification program was (and still is) also available to civilians.
Regards Hardy