For a C-Spine injury, it is imperative that the head is immobilized. A back board (can be a challenge to improvise one) is needed to ensure the head and back stay put while extricating. Keep the helmet on if possible. The helmet is easier to hold that the head and lifts the head a bit. It also makes it easier to secure the head so it doesn't move sideways. Once the victim is in a safe location, either on shore or perhaps on a mid-stream flat rock - wait for professional help if possible. They have the special training and certainly the right equipment like a real back board or litter. Always seek help if possible for these types of injuries - why take chances. The following EMT video, especially the second half is a great demonstration on the technique and all of the precautions you need to take: C-Spine Immobilization.
One way to flip an unconscious swimmer safely face up are using both his arms to cradle the head and carefully rotate. The other way is the manually immobilize the head and slowly duck under them. Administer CPR as soon as possible if necessary, don't wait to get them on shore. Remember your a team, have someone heading off to get professional help quickly. Here is a great article with diagrams on handling C-Spine injuries in aquatic environments: Handling C-Spine Injuries.
Here is a video from a Safety & rescue class practicing a C-Spine Rescue: C-Spine Rescue Practice Video.