Learn German: Der Teppich gehört sofort gestaubsaugt. - The carpet needs to be vacuumed immediately.

Using gehören + a past participle (here, gestaubsaugt) is a passive alternative that expresses strong necessity or obligation. It means exactly the same thing as the modal passive construction Der Teppich muss sofort gestaubsaugt werden, but it is more colloquial and very common in everyday spoken German (especially in southern Germany and Austria).

How can I tell when gehören means "needs to be" rather than "belongs to"?

Look at the end of the clause! If gehören means "to belong to," it is usually paired with a dative object (e.g., Der Teppich gehört mir - The carpet belongs to me). When it is used alongside a past participle at the end of the sentence, it triggers this "necessity passive" meaning instead.

Why is the past participle gestaubsaugt? Shouldn't it be a separable verb, making it staubgesaugt?

You would naturally think so, since it's formed from a noun (Staub) and a verb (saugen). However, staubsaugen is treated as an inseparable, weak verb in standard modern German. Because it is inseparable, the ge- prefix goes at the very beginning of the word, resulting in gestaubsaugt.

Do I ever need to add zu when using gehören as a passive alternative?