That is an interesting question Jean. It seems that we are used to and familiar with the kogai on the omote and the kozuka on the ura side of the koshirae. The origins of this convention seem to be unknown and from what I have read it emerged during the Edo period.
I have gone through my koshirae references and I can find only one koshirae that has only a kogai mounted on the reverse (ura). It is illustrated in "Uchigatana" from the Tokyo National Museum and is a wakizashi of a daisho owned by Torii Mototada (son of Hideyoshi) who lived from around 1567 to 1688. I can`t get any pics up of it but it is number 46 for those who have the book. This probably is not the only existing example but it seems to demonstrate that they are quite rare for whatever reason.
The Tosogu no Kigen discusses at length the origins of the kogai. It mentions that before the Muromachi period and the emergence of the uchigatana that there are records of koshigatana and the like being mounted with kogai in the position of where we usually find the kozuka. Also it mentions that the kogai is much older than the kozuka which I read as some people think kogai once existed before kozuka were invented.
Concerning the uchigatana and kogai, there are theories that Tosogu no Kigen cites "On swords worn in camps when on campaign, only kogatana are mounted, but in camps at the castles, only kogai are worn and this is on the omote as well as on the ura side"
There exists kogai where the motif is orientated in the same way as on a kozuka. For certain designs that have a top and bottom, the orientation of the design on some kogai make them suitable only to be worn on the ura side of the koshirae.
So to answer your question, I think that from the Edo period onwards, the kogai would have been placed on the omote (outerside) but this probably was not always the case, especially before the Muromachi period and the advent of the uchigatana.
I hope these ramblings answer you question and I also hope that someone with more knowledge than me can add something to this discussion.