My father was assigned to the old Sheepshead Bay firehouse for Engine 246, Engine 327 and Ladder 169. He used to remember how good it was getting a job and having two engines arriving together and being able to get two lines in operation so quickly. He would remember that when a box was received, E 246 and E 327 would typically rotate who left quarters first. If the location was to the right of quarters, E 246 left quarters first. If it was to the left of quarters, E 327, in their large Ahrens Fox pumper, left first. I don't know if that reflected assignment cards but it was practical.
Things were different in firehouses with extra companies.
Having 3 or 4 companies, and sometimes a chief or special unit, in the same quarters made many firehouses cramped - lockers, beds, offices, gyms, TV rooms, kitchens. The apparatus floor and outside parking lots were filled. Especially older and smaller firehouses. Meals were big. Apparatus often had to be moved to let rigs parked behind on the apparatus floor respond.