What it is ‘about you’ (interesting that you used that phrase instead of ‘in you’), that is carried from one rebirth to the next must be sanskars. mental traits and the mind stream. The mind stream must be reborn otherwise it would not be possible for some people to recall past lives. ‘We’ (who’s ‘we’?) take our karma with us (or our karma takes us along with it), but karma determines our rebirth and influences what happens to us subsequently rather than being an inherent characteristic.

To say that something lacks ‘inherent’ existence is not to say that it doesn’t exist at all, which, correct me if I’m wrong, your question seems to assume.  It merely means that something does not exist from its own side, that is, it is not permanent, unchanging and the same from whatever viewpoint it is looked at. I could go into analogies and further explanation but I won’t.  You should be able to come up with them yourself.

Exactamento (excuse my Welsh).

To go back on what I’ve just said and at the risk of descending into philosophical debate on an anti-or a-rational Zen forum, inherent existence is not a thing but a concept. The concept being that there is a thing which has inherent existence, is permanent and unchanging and indivisible and has exactly the same properties and appearance and essence etc no matter from what point of view it is looked at or even thought about.  In philosophy this thing is sometimes given the name the Thing-in-Itself. Other names for it are the Absolute or God.  Given that we humans, along with all other things, are not the Absolute or God (though some of us may think we are or maybe a part of it), we are said to lack inherent existence.  However: this does not mean – is not the same thing as saying – that we don’t exist at all. 

So there shouldn’t be any problem accepting that there is ‘something’ which is carried from one rebirth to the next any more than there is a problem accepting that it is possible for a person to stand at the top of one hill and see the surrounding landscape from one viewpoint and then half an hour later walk to the top of the next hill and see it from another.

Darn it! I’ve gone into an analogy. I never keep my word.