If I am going to carry the deer out I gut it , make it into a pack and carry it out , If it is too far to carry the whole deer I remove the back straps , hind legs and fore quarters , bag them in old pillow cases , put them in my pack and carry them out , Heart ,Liver, are checked for disease and the Kidneys and tongue are also salvaged .
I have seen many pictures of North American hunters hanging deer by the head , here in New Zealand they are always hung head down , by the hind legs . This is especially so if the animal is hung in a tree to be gutted or is gutted on a steep slope , the offal falls forward and out of the gut cavity and in the unlikely event of a broken gut , contamination of the meat is only on the lesser forward cuts .
I have seen many pictures of shot deer and quite honestly the gut cavity often looks like it has been opened up with a hedge trimmer .
The way to cleanly gut a deer is to first make a circular cut around the anus and pull it out a little way then tie it off with a string then push it back in
Make a small incision with your knife just below the breast bone , then with your knife blade pointing up place it between your two fingers of your left hand and slide your hand , palm up , and the knife down the center of the belly towards the legs . This keeps the knife blade cutting the skin from inside, and the back of your left hand pushes the gut away from the knife tip. The tip being between your fingers keeps it from puncturing the gut, while the blade cutting upwards from the inside will not be blunted by hair or drop cut hair all over the meat . Cutting into deer hair is a sure way to blunt a knife
When the gut bag drops out the intestine is pulled with it and the anus with the string tied around it comes out cleanly without dropping poop on the meat .
The thing to remember when you are shooting , gutting and processing a deer you are making it into food and cleanliness is paramount .
PS If you are going to cut a Stag's throat while it still shows signs of life just remember they can still horn you or kick and maybe connect with your knife