It isn’t our feelings that are relevant; it’s canon law. Under canon 1374 of the 1983 Code of Canon Law, Catholics are forbidden to join societies which plot against the Church. The Congregation for the Doctrine of the Faith applied this to Masons, indicating that Catholics who join Masonic organizations are engaged in serious sin and are to be barred from the Eucharist.
The Church has also judged that the Odd Fellows, the Knights of Pythias, the Orangemen, the Sons of Temperance, and the Communist Party are forbidden societies. Most Americans are familiar with the Masons and the Communist Party, but less familiar with the others.
The Odd Fellows was formed in England in 1812 and was brought to America in 1819. Like the Masons, it is a quasi-religious society, which is one of the Church’s chief objections against it. The group has chaplains, altars, high priests, ritual worship, and funeral ceremonies. Also like the Masons, its members are indifferentists and teach the equality of Christianity, Judaism, and Islam before God.
The Knights of Pythias was founded in 1864 by a group of Masons and suffers from many of the same problems as the Masons and the Odd Fellows. This organization places emphasis on the pagans Pythagoras, Damon, Pythias, and Dionysius as teachers and models for moral life.
The Orangemen are militaristic Irish Protestants who are intensely anti-Catholic and are responsible for much of the anti-Catholic violence in Northern Ireland.
The Sons of Temperance was founded in New York in 1842 and is far more than just a temperance society (which the Catholic Church has nothing against; there were many Catholic temperance societies in 19th-century America). Like the Masons, the Sons is a quasi-religious society with its own religious rituals, but it admits both men and women to its lodges instead of having a parallel organization for women as do the Masons and most similar groups.