When we think of compassion we think of sympathising with those who are suffering in some way. This relationship with the person who is suffering is one with a them and a me, which has an innate division. In Buddhism compassion is very important, as is the concept of emptiness, which is essentially that all things are empty of labels, characteristics and preferences, to put it simply.
Compassion, in its best form, is one where there is no division between yourself and others; there is an innate unity. This would mean that the compassion you feel is for the suffering of another living being and the wish to end this suffering, because you see no difference between them and you. This is the kind of compassion I would advocate for, the kind that has no divisions, only an understanding of the suffering of others and the wish to end it.