There is a convention that, where there is no evidence to the contrary, the owner of a fence (and, consequently, the person responsible for maintaining it) is regarded as being the householder who has the fence posts nearest to his property.
This convention has been recognised by the courts in some specific cases but it doesn't form part of statute law.
Which side the posts are on means nothing. Any liability to maintain a fence would be stated in the Deeds/Registration, but this would be very, very unusual. Mostly there is no statement about fence maintainance so consequently there is no liability upon anyone to maintain a fence. The ownership of a fence will be marked upon the Deeds/Registration with a T, but this does not mean that the owner is either liable to maintain it or to replace it in whole or in part.
I always thought that with adjoining properties there was a sort of rule that I would be responsible for the fence on the left hand side as I looked out of my front door?
Fences are often not marked on the Deeds with a T. If they are not, the convention Buenchico mentions usually applies unless there is evidence to the contrary. While there is no legal liability to maintain the fence, when you sell one of the questions on the pre-contract enquiry form will ask which fences you consider you are responsible for, so it is as well to know this.