Today there arose an experience of a lovely little bit of Catholic tradition. I and two other friars were praying the sacrament of Anointing of the Sick for a certain priest. There is a striking little difference when a priest receives this sacrament, explicit in the former rubrics of Extreme Unction and still practiced, though missing from the current rite.
Everyone else is anointed on the palms, while priests are anointed on the backs of the hands.
The palms of a priest's hands already bear an anointing, which he received at his priestly ordination. Because this anointing remains even long after the material, visible signs of it may have been washed away--see this favorite old post for a reflection on that--the palms of the hands are not anointed again.