Place the corned beef and broth in a Dutch oven. Add the pickling spices and garlic and bring to a boil over high heat. Reduce heat to low, cover, and simmer until the beef is cooked, about 3 to 3 1/2 hours.
Add the potatoes, carrots and turnips to the pot and cook until the meat and vegetables are tender, about 30 to 45 minutes. Meanwhile, cook the beets in a separate saucepan of boiling, salted water for 30 to 45 minutes (this prevents the beets from discoloring the other vegetables).
Remove the beef and all of the vegetables from their broth and arrange on a heated serving platter. Pass the mustard and horseradish, if desired.