What we perceive of as left vs. right seems to be rooted in hashkafic differences and different views of how mekorot are to be applied in the 21st century.
One cannot draw lines in the sand so easily, however. My reading of sources, and my choice of rabbis lead me to positions which differ along left-right lines. Nobody is left-right all the time. Example- Rav Benzion Uziel, zt”l, was someone who paskened very courageously on issues such as conversion, agunot, women’s place in society, abortion, etc. Yet he didn’t believe modern scientific knowledge was of much importance and he seriously believed that talmudic accounts of parthenogenesis should be taken seriously.
Likewise, I know a rabbi who for many years was the rav of a traditional/conservadoxish shul, yet he refused to carry in his town’s eruv, because this rabbi is someone who is a Rambamist/legal positivist and not a legal realist. Plenty of people in all sorts of old world levush, on the contrary, will carry in eruvin with very scant halakhic basis.
Rav Yitzchak Abadi has stances on kashrut that most MO people wouldn’t hold by, yet he is extremely black hat and says that shaking a woman’s hand under any circumstance is yehareg ve al ya’avor.
I think such matters should be looked at on a one-by-one basis. It’s unfair to make blanket generalizations about people.
I will say, however, that MO tends to be more of a sociological categorization, in the sense that people will assume someone’s MO status based on appearances, or even laziness in observance, such as not saying asher yatzar after using the restroom. I think many MO people view halakha and observance in terms of minimalist floors, rather than maximalist ceilings; i.e. they believe that keeping shabbat, eating kosher, going to shul once a week, and keeping taharat hamishpacha are sufficient to live Jewishly.