In the binomial pricing model, why do the magnitude of the up factor $(u)$ and down factor $(d)$ have to be multiplicative inverses? I have read from multiple sources that the reason for this is that an up move followed by a down move $(ud)$ will have the same effect as a down move followed by an up move $(du)$, which simplifies calculations. However, doesn't this property still hold when $u$ and $d$ are not inverses? For example, $1.1*.8 = .8*1.1$
It does not need to be so always. You can always relax that assumption and come with the pricing by using the fundamental principles. As @Kermittfrog and @Dimitri Vulis commented it is just a matter of convenience for calculations and is called the recombining property. You can find an example in this link here which does not use this assumption to price the options.