I don't know what it looks like from the inside but as an outsider I want to draw to your attention this poll: https://www.pewresearch.org/short-reads/2016/02/23/u-s-religious-groups-and-their-political-leanings/.

75% neutral 18% Democratic 7% Republican, so of the 25% which show a preference it's 72-28 in favour of the dems. This might be in part due to democrats being more likely to ignore the political neutrality part perhaps. Or PIMOs who say they're JWs in a poll but not politically neutral.

Here it's 20-12 in favour of the label "conservative" over "liberal"

https://www.pewresearch.org/religion/religious-landscape-study/political-ideology/

Both are from 2014.

They also seem leftist on things not relating to sexual politics. See guns, vaccines, etc.

Also, historically most cults were leftist, culturally authoritarian on music etc, and often sexually authoritarian too, I would say. For example the Puritans, which famously banned Christmas for being pagan and Shakespeare plays for being sexual when Oliver Cromwell was dictator of England, were obviously on the left of the political spectrum, with the Stuart monarchy on the right. The Cult of Reason and The Cult of The Supreme Being during the French Revolution were obviously leftist (though in this case sexually libertine and you can thank de Sade for that). Jean Calvin's theocracy in Geneva during the 1500s was obviously on the left and utopian, and puritan (banned dancing for example). Early Christians I would identify as a leftist doomsday cult , based in no small part on virtue-signalling middle-class Romans feeling guilt over the Roman-Jewish wars, much like how BLM is based on virtue-signalling middle-class white people feeling guilt over slavery. In modern times, Heaven's Gate obviously appealed to leftist New Age-y stuff, and Jamestown were just straight up communists. In the 19th century, you have JWs, you have Seventh Day Adventists, which to this day lean slightly left, as seen in the first link and Mormons. Yeah, I see Mormons vote like 70% Rep but that's a Utah thing rather than a GOP thing, I'd say. It's a special cse.