"They reported being told to..." Trying to explain

Solution 1:

In comments, John Lawler wrote:

Yes, it's grammatical, and so is They reported having been told to... The role of being (and of been in the one above) is to be the auxiliary verb for the Passive construction, which consists of some form of the auxiliary verb be followed immediately by the past participle of some transitive verb (here it's told, the past participle of tell). The reason why -ing forms are OK after report is because report allows a gerund complement. Every verb has its own rules about stuff like that.

There are four major types of complement clause, two finite and two non-finite. The non-finite ones are the gerund and the infinitive complement types.