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Cut the meat and fat into chunks that will fit into your meat grinder. Combine the salt, sugar and all the dry spices (except for the achiote paste) with the meat and fat, mix well with your hands and chill it until it's almost frozen by putting it in the freezer for an hour or so.
Take out some hog casings and set in a bowl of very warm water. Mix the ice water with the achiote paste and chill it in the fridge.
Grind the meat mixture through your meat grinder (you can use a food processor in a pinch, but you will not get a fine texture), using the fine die, about 4.5 mm. If your room is warm, set the bowl for the ground meat into another bowl of ice to keep it cold. Make sure the meat mixture is very cold before moving on to the next step: You want it between 27°F and 35°F.
Add the water-achiote mixture and mix thoroughly either using a Kitchenaid on low for 60 to 90 seconds or with your (very clean) hands. This is important to get the sausage to bind properly. Once it is mixed well, put it back in the fridge and clean up.
Stuff the sausage into the casings all at once. Twist off links by pinching the sausage down and twisting it around several times; do every other link and you will only have to twist in one direction. Or you could tie them off with butcher's string.
Hang the sausages in a cool place for up to overnight (the colder it is, the longer you can hang them). If it is warm out, hang for just one hour. Once they have dried a bit, put in the fridge until needed. They will keep for at least a week in the fridge. If you are freezing the sausages, wait a day before doing so. This will tighten up the sausages and help them keep their shape in the deep-freeze.