The image from ANNUAIRE du COMMERCE DIDOT-BOTTIN 1922 Paris shows that the ROTO logo is a mark for the Belgian publisher 'ROTOBROME'. I think ROTOBROME is actually a trade name as www.cartophilie.be identifies the ROTO mark as belonging to the company Rotographie Belge Bruxelles. This was probably related to the German company Rotophot as
it was a member of the NBC cartel.
www.cartophile.be also shows an undated advert for "ROTOBROME", Société Belge d'Impression Photomécanique, 95-97 Avenue du Pont-de-Luttre, Forest-Bruxelles, "Seule usine de ce genre en Belgique". An entry of the Brussels Bottin of 1920 is Cesar J, chaussée de Waterloo 294.
Cards sold in Britain had the word "Rotobrome" and no logo." Cards usually say "Printed in Begium" or equivalent.
The ROTO mark is sometimes surmounted by a cockerell.
I'm guessing that the other Belgian cards shown here with R,T,B in their logos are from the same source. Card RTB 3438 below could be interpreted as "RTR" but sometimes little o's appear between the letters in this form.
Cards with various RB logos I show separately, though these may be the same as well.
from ANNUAIRE du COMMERCE DIDOT-BOTTIN 1922
Roto 6374
RotoB 5477
RTB 3438