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Cut the pork shoulder into chunks that will fit into your grinder. Mix this with the salt, chopped garlic, pepper and cumin, and refrigerate overnight, or up to 2 days. You can skip this step, but your sausage will not bind as well.
Get out about 15 feet of sheep casings and soak them in warm water. If you want, flush them with water; this helps the stuffing process and will let you know if you have any leaks.
Grind the meat through a medium (6.5 mm) die. Chill the mixture in the freezer until it is about 33 degrees Fahrenheit.
When the mixture is cold enough, mix the achiote paste and the orange juice together, then mix that into the meat along with the carrot fiber, if using. Mix well with your (very clean) hands for 60 to 90 seconds. Your hands will ache with cold. You'll know the sausage is ready when it coheres in one mass. You'll also start to see whitish streaks form on the side of the container you're mixing in. You can also do this mixing in a stand mixer on low.
Pack your sausage into a stuffer and thread on a length of casing. Slowly ratchet down the meat to remove all air from the stuffer and the tube the casing is on. Leave about 4 inches of casing off the end of the tube; you'll use this to tie off later.
Stuff about 2 feet of sausage rather loosely, leaving a couple inches of casing on either end. Do this with all the sausage and casings. When you have all the sausage in these lengths, gently compress each one looking for air pockets. Use a needle or sausage pricker to pop any air pockets. Gently compress the links to fill that gap. This takes finesse not to burst the casing.
When all you links are ready, tie the ends to each other to form an elongated loop. Hang your longaniza to dry. At room temperature, hang an hour or two. Ideally, you'll hang links between 33 and 39 degrees Fahrenheit for 24 hours, or even up to 3 days. Barring that, leave them uncovered in the fridge overnight before eating or freezing.