Moses wore a veil over his face after Sinai, because the shining of his skin frightened the people (Exodus 34:30). But there was one moment he always took it off. Targum Pseudo-Jonathan on (Exodus 34:34) preserves the rhythm: when Mosheh went in before the Lord to speak with Him, he removed the veil from his countenance until he came forth.
The Targum's small detail carries a huge idea. Before God, no covering. Before Israel, a veil. The human intermediary dresses down only for the divine audience, and puts the covering back on for the public one.
The rabbis drew out two meanings. First, practical: you cannot mediate with a screen between yourself and the One you are mediating with. The prophet must meet God face-exposed, or the channel is blocked. Second, theological: Moses' face was itself a borrowed light, and that light had to be returned to its source each time the prophet came back into the tent. Removing the veil was not display, it was re-charging.
There is also a poignancy here. Moses lived in two modes after Sinai. Publicly he was muffled, dimmed, kept at a reverent distance by his own people. Privately, in the Tent of Meeting, he was fully himself, fully lit, fully near. The veil was the cost of prophecy exercised in public. The unveiling was the prophet's private life.
The takeaway: the truest self is the one that stands before God without covering. Everything else, the veil, the distance, the careful titration of light, is a concession to the world's capacity to bear what a prophet has seen.