自性 is often misunderstood to be an independent, unchanging Self
similar to Hinduism's atman-brahman.
However, if understood correctly, it is simply the mind's luminous
and empty nature.
You want to call it Mind, Nature of Mind, Buddha-nature,
Self-nature, Thusness, Suchness, Dzogchen, Mahamudra, etc, it's all
fine.
http://awakeningtoreality.blogspot.sg/2007/03/self-liberation-through-seeing-with.html
6.
As for this sparkling awareness, which is called "mind,"
Even though one says that it exists, it does not actually
exist.
(On the other hand) as a source, it is the origin of the diversity
of all the bliss of Nirvana and all of the sorrow of Samsara.
And as for it¡¯s being something desirable; it is cherished alike
in the Eleven Vehicles.
With respect to its having a name, the various names that are
applied to it are inconceivable (in their numbers).
Some call it "the nature of the mind" or "mind itself."
Some Tirthikas call it by the name Atman or "the Self."
The Sravakas call it the doctrine of Anatman or "the absence of a
self."
The Chittamatrins call it by the name Chitta or "the Mind."
Some call it the Praj?¨¢p¨¢ramit¨¢ or "the Perfection of
Wisdom."
Some call it the name Tathagata-garbha or "the embryo of
Buddhahood."
Some call it by the name Mahamudra or "the Great Symbol."
Some call it by the name "the Unique Sphere."
Some call it by the name Dharmadhatu or "the dimension of
Reality."
Some call it by the name Alaya or "the basis of everything."
And some simply call it by the name "ordinary awareness."