I remember a bit of folklore that suggested it was more hygienic to put the lid (that's the lid on the seat) down after use. Is this the case?
Yes, if you close the lid before you flush.
This topic was report by Karl S. Kruszelnicki, a noted skeptic, science reporter and popular-science author in Australia. He cited Dr. Charles Gerba, a microbiologist from the University of Arizona, who has published on the topic.
If you flush with the lid up, a polluted plume of bacteria and water vapour erupts out of the flushing toilet bowl.
Unfortunately, Dr Karl (as he is known) did not provide a detailed reference.