When both sternocleidomastoid muscles work will they flex or extend the neck and head? I thought it would extend the head backwards since the mastoid process is behind the axis but different sources say different things...

I think your problem is not necessarily on the action of the muscle but on the physical actions defined by the terms flex/extend the head and neck. If I remember my anatomy lessons over 20 years ago you need to consider the movement of the head and the neck separately ( ie it flexes the neck and extends the head) - you might also wish to consider the older terminology such as the fact that it "protracts the head" and the annoying fact that it also extends the neck iff the neck is already extended.

 

SCM is an object lesson in operative anatomy - that is to say that muscles act in harmonious groups that act and counteract together; very few joints are simple hinges that are flexed and extended.