What is the quality of being concerned or apathetic, worded in a neutral term?

For example - the quality of "prettiness" and "ugliness" shouldn't be either but "appearance" as one can be pretty or ugly, but not "appearanced".

On the same weight, the quality of "interest" and "apathy" shouldn't be either. It should be a word neutral to both of ends of it's own dichotomy. Concern, curiosity, or interest do not seem to cut it for me.

Solution 1:

After sleeping on this I suggest awareness as a possibility that is derived from the adjective aware.

At one end of the spectrum we have phrases relating to strong awareness (= interest) discussed in

EnglishStackExchange

such as:

keenly aware, painfully aware, acutely aware

At the other end we have examples relating to weak awareness (=apathy or indifference) such as T S Eliot's

Goodreads

“She turns and looks a moment in the glass,

Hardly aware of her departed lover ..."

and

New York Times

So ignorant was I of New York real estate that I was barely aware of the Great Landlord-Tenant War of 1997, which was then raging.

This neutral range of meanings is reflected in Merriam Webster's account of awareness:

Merriam Webster

awareness:

the quality or state of being aware : knowledge and understanding that something is happening or exists

promoting a heightened awareness of the problem

seemed to have only a slight awareness of what was going on

an acute awareness of subtle differences

They hope to raise awareness of endangered species, encouraging donations and attracting tourists to the region to support conservation efforts. (New Scientist)

The alternative press played a particularly important role in raising political awareness and facilitating mobilization against the regime in South Africa … (Larry Diamond)

I also considered involvement but it does not seem to match your requirement of wide range and bipolar neutrality so well.