The order of the electrical crossover slope is simply the number of branches in the circuit. A first order has one positive branch (a coil for the woof, a cap for the tweet), a second order adds a branch to ground (coil for the tweet, cap for the woof), and so on. But the performance characteristics of a speaker depend on the order of the acoustic slopes--the rate at which the woofer actually rolls off at the top end, and the tweeter at the bottom. That's what the ear hears, and what affects its lobing pattern and phase characteristics. That will depend not only on the electrical circuit--the roll-off that arrives at the speaker's input terminals (also known as the transfer function)l--but on the inherent response of the woofer (and on the shape of its impedance curve). If the woofer has a sharply rising response above the intended crossover point, you could end up with less than a 4th order slope. If the driver is already starting to roll off at the crossover point, you could end up with a steeper than 4th order slope. In general, but not always, you end up with a steeper slope than the order of the electrical circuit. So it's not unusual for a speaker with a 4th order acoustic Linkwitz-Riley crossover to have a 3rd order electrical circuit, or frequently a second order. It's quite unusual for the electrical and acoustic slopes to be equal. At the extreme, if a woofer and a tweeter had very smooth 4th order roll-offs on their own, you wouldn't need any crossover at all (ignoring baffle step compensation), and the electrical "circuit" would be of zero order.
