I think that, while it’s not possible to draw hard, exclusive lines here, progressive sensibilities evince a gravitational pull toward utilitarianism and conservative instincts gravitate toward deontology. (Both sides can care about virtue, and for both, I’ll conjecture, it will be connected to the larger frameworks I’m talking about in this post.)
I’m not saying there are no people on the left who care about moral rules; clearly, many do. But it is possible to be a perfectly legitimate leftist who thinks all moral rules can be justifiably trampled in the quest for utopia.
The leftist wants good things for all, with the smallest number possible excluded (preferably none at all), and to have the benefits as widely distributed as possible without an advantaged, hoarding class taking too much of what could be shared.
Some people on the right sincerely work toward a similar future, but it is possible to be a genuine conservative who doesn’t particularly care about child labourers, or slavery, or needless deaths or disability, etc., so long as the superior people at least (whether that’s the capitalists or the warriors or the philosophers etc) are doing the things they are obligated to do.
Often you’ll hear people on the right arguing against the left as if they were utilitarians: “Well actually though, if you really did care about all those people doing well, then you should be focused on economic liberty / law and order / entrepreneurial spirit / virtue / etc., and I can prove it.” But in that case, the blessings that stretch out to fill the ends of the earth are the effect of doing the right things, and therefore are the proof of their rightness. The greatest good of the greatest number isn’t the goal, it’s, at best, one of the byproducts of working at the correct goal.
One of the biggest differences between right and left, in theory, tends to be the attitude toward hierarchy. The left wants to flatten the hierarchy out as much as possible, and the right is at least indifferent (or often enthusiastic) about one hierarchy or another rising up within a population. There are a few different ways we could relate that difference to the two moral approaches I outlined above, and I’m not sure which one is the main reason, but I definitely think they are related.
How do you adjudicate between the two sides, as if we had a neutral moral standard to use? And, if you wished to fuse the two into a single whole, as is my first impulse, we are left with the question of which one will be the graft and which will be the host. Difficult conundrums.