The conspiracy trope is that the organization is secretive and little-known. They work together toward an end which they could not achieve under the bright light of day, so closely huddled as to share breath, so let's work from that.

It's certainly true that politicians work together to achieve shared goals and these goals may be wildly unpopular. For example, if pols think that a very popular piece of legislation admired by a large section of the country ought to go, they might cook up a repeal bill in the dead of night and have it on the floor for a vote the next business day. They might even submit one bill and then alter it to gain supporters and then pretend they hadn't done any such thing. Other politicians might then ask for a delay in the proceedings, because they smell a rat, and use that time to write a public letter excoriating their opposites for all that skullduggery. Both of these essentially happened with the Kansas-Nebraska Act, with our first cabal being essentially Stephen Douglas and a group of Senators who shared a boarding house on F Street working in conjunction a lone Whig Senator who outmaneuvered them and then had to be brought on board during an extended carriage ride around Washington City. Their opposites, antislavery politicians, pulled the delay and publish gambit in response. Separate from them, there is a not especially credible story that antislavery Senator William Seward put his Whig pal up to outmaneuvering the F Streeters and this started the fall of dominoes toward Sumter.

Are the F Streeters, their antislavery foes, or Seward playing the role of a shadowy cabal? Not really. They're all public men of very well-known affiliations. Every one of them is a United States Senator with pretty clear positions on slavery's expansion, the issue that made the Kansas-Nebraska Act so odious. Nor are they engaging in some kind of secret favor-trading to make this all work out, but rather doing normal politics of coalition-building and vote counting. Likewise their antislavery foes may have stopped debate for a few days under false pretenses (claiming they needed to read the bill over first when really they were writing a public letter to raise the alarm) but that's not exactly cackling in your volcano lair stuff. By definition, a conspiracy is not public politics as normal and both of these situations are exactly that.

That doesn't mean that people didn't view the whole business as shady, though. In the North, the continued ability of the slave states to enact expansions of slavery against the will of the more populous section is often viewed as a kind of conspiracy for the Slave Power...but also the Slave Power is understood as a kind of formal political power embodied by sectional parity in the Senate, extra seats for enslavers in the House, the consequent arrangement of the electoral college, and the real fact of consistent proslavery administrations from Washington until Lincoln. There may have been something conspiratorial about that thought, but the favor trading and so forth that secures Yankee votes for slavery all happens in public, whether by means of concessions granted or patronage distributed.

We see similar things with every sectional clash going back to Philadelphia. Slavery is always the top priority for white Southerners but is rarely so for Northerners, so the South gets its way. Even as the North eclipses the South in population, and so the House, and much later still theoretically the Senate, the South's greater sectional unity wins out right through 1860. The arrangements include private talks and negotiations, as such things always do, but you can't get much farther out of the shadows then elected officials carrying out their stated policy preferences.