The Leviathan Cross is sometimes referred to as Crux Satanus, which means the "Satan Cross". Depicted on the bottom is an infinity sign (∞), and above is a double cross (‡). The Double cross symbolizes protection and balance between male and female. The infinity sign most likely means eternal rest/happiness, this may have been used by Anton Lavey in The Satanic Bible to represent that Hell, the kingdom of Satan, was actually an afterlife of pleasure, balance, and truth, while Christianity is a religion based off of corruption and lies. This is what the ideology of Anton Lavey and the followers of the Church of Satan believe, and this is what this symbol represents.
An upside-down crucifix, however, is not a symbol of Satanism. That would be referred to as "The Cross of St. Peter" which was a key symbol to the religion of Lutheranism, which is a sub-religion of Christianity.
In Alchemy, the Leviathan's cross is a symbol for sulfur, which is one the three essential elements of nature.
The origins of the Leviathan Cross is created/used by The Knights Templar. The symbols was then adopted by Anton LaVey, founder of the church of Satan. Previous associations with the cross and satanic foundations are unable to be identified.