"this invisible government war party goes back perhaps to the 1846 Mexican war..."
Are they still using the secret handshake?
Tarpley believes the Illuminati are basically a fiction furthered by the British empire to distract from their own activities. That's why they chose a supposedly German outfit to blame, as Germany was their great rival. So I don't know about any handshakes. However, the structure of secret societies like the Masons can be used for all kinds of purposes, e.g. the P2 lodge.
It's hard to find a term for this phenomenon. Invisible government sounds too spooky. But then, a lot of them are "spooks" no doubt.
As for the date 1846, actually the techniques of fake terror and other forms of provocation go back almost to ancient times, as far as I know. Tarpley cites the Guy Fawkes plot of 1605 as the fake terror act that helped found the British empire by inciting to war against Spain. He says the British learned the stuff from the Venetian oligarchy, who moved there around that time. They got it from the Byzantines.
So 1846 is not that long ago in historical terms.
Another suspicious provocation around the same era was the the attack on Ft. Sumter, the suspects being the British or their agents of influence.
One of the fascinating things is that the Venetians allegedly used religion as one of their favorite means of fomenting strife in their divide and conquer strategy. All those religious wars in the 17th century. Sound familiar today? Look at what we got now - religious and factional warfare in the middle east.