Learn Dutch: We hebben structuur nodig om te mogen beginnen. - We need structure to be allowed to begin.

In an om ... te clause with multiple infinitives, the te must go immediately before the first infinitive in the cluster. Because the modal verb mogen comes before the main verb beginnen, the te locks into place right before mogen.

Why is nodig placed at the very end of the main clause?

The phrase nodig hebben (to need) acts similarly to a separable verb. In a main clause, the conjugated verb (hebben) takes the second position, and the separated part (nodig) gets pushed to the end of that clause, right before the om ... te structure begins.

Could we swap the verb order and say om te beginnen mogen?

No. When you have a cluster of infinitives at the end of a clause, the modal verb (mogen) almost always comes before the main verb (beginnen). Therefore, the correct order is mogen beginnen.

Why use mogen here instead of kunnen?

Mogen specifically expresses permission (to be allowed to), whereas kunnen expresses ability (to be able to). This sentence implies that having structure is a strict condition that must be met before someone gives permission to begin.