As far as I can tell there is two issues at work here:

1. Can an angel lock itself by looking at part of itself, say an arm or a hand?

Clearly not, as pointed out, they cover their eyes, which would defacto risk locking themselves. Given that they lock instantly, chances are they cover their eyes a fair chunk of the time. The nature of the Angles is to be observed, if you get existential, the Angel already "knows" of them-self can thus can not observe it in the same way as an outside creature.

2. Can an angle lock by looking in the mirror.

In this case yes, the image of an Angel becomes an Angel, and thus observe each other. Possibly locking each other.

One could argue that the image is an EXTENSION of the original angel, but evidence points to counting as a separate entity (although the angels do have some signs of telepathic or hive-mind properties)

As far as I can tell, there are two issues at work here:

1. Can an angel lock itself by looking at part of itself, say an arm or a hand?

Clearly not, as pointed out, they cover their eyes, which would de facto risk locking themselves. Given that they lock instantly, chances are they cover their eyes a fair chunk of the time. The nature of the Angles is to be observed, if you get existential, the Angel already "knows" of them-self can thus can not observe it in the same way as an outside creature.

2. Can an angle lock by looking in the mirror.

In this case yes, the image of an Angel becomes an Angel, and thus observe each other. Possibly locking each other.

One could argue that the image is an EXTENSION of the original angel, but evidence points to counting as a separate entity (although the angels do have some signs of telepathic or hive-mind properties)