Which political party did the majority of the Mormons belong to in the 1860s?
I don't know of any official statistics, but one can assume that they were Independents. They had been harassed by both Democratic and Republican governments in Washington D.C. and probably wanted nothing to do with either party at that time. Joseph Smith ran for President of the United States under an Independent platform in the 1844 election, and Brigham Young ran for territorial governor of Utah apparently without stating a party and received 100% of the 10,000 votes.
They were Anti-Slavery, Pro-State's Rights, Pro-Women's Sufferage, and deeply rooted in the defence of the Constitution and the Founding Fathers.
In the 1860's, the majority of members of the Church of Jesus Christ of Latter-day Saints (the "Mormon" church) lived in Utah or other western states and territories. There were some members scattered throughout the United States, Canada, the Pacific Islands, and Europe, but Mormons at that time were encouraged to move to Utah.
Up until the 1850's and 1860's Japan had a political system similar to that of medieval Europe but did not keep this when it made contact with Western governments and later changed to a Democracy as a result of the American occupation of Japan after WWII. Today it has a made special democracy for itself by mixing Japanese and American cultures and ideas .