Congress and President have, as I understand it, a mutual veto - but in the case where both are controlled by the same party, they tend to defer to the party organisation, which in turn defers to the President. Where they don't, you get shutdowns.
President tends to dictate congress when they are popular and congress is of the same party. While there can be some push back, it will become costly come election time. SCOTUS has strongly defined limits on what they can do, but given they are made up of picks by the President and approved by congress the shape of the SCOTUS changes depending upon whom is in power when the existing members leave.
The President is generally immune to congress until they do something that gets enough congressional support for impeachment. This has proven extremely hard to do so far. The President also gets significant say over any issue that doesn't strongly unify congress.
Thus, for an issue that is popular in congress and unpopular with the President, but not popular enough in congress to override a veto, the hierarchy does President > Congress (a > b means a is above b). SCOTUS almost never gets involved when it comes to passing laws.
When it comes to an issue that is popular enough congress is willing to override a veto and which is popular enough with voters that congress going against the President on it isn't a threat, then congress > President.
I don't think anyone is claiming there is a simple hierarchy that consistently applies regardless of scenario. Even in a classical monarchy the hierarchy can shift for extreme enough scenarios.
There are very clear hierarchies in Congress (Speaker/Majority Leader, Majority Whip, Committee Chairs) and White House (President, Vice President, Cabinet). There is no power structure in SCOTUS: each of the justices is equal and has nobody to answer to.
There are likely shadow organizations in the Congress and White House with their own (unpublished) hierarchy, but probably not in the SCOTUS due to its size and the implicit potency inherent in being a Justice.
SCOTUS is a nearly-pure democracy. Every issue of substance is put to a vote, and everyone has freedom to act individually in anything not put to vote, along with ample funding for their desires (law clerks for work, and indivual pay for personal perks). (Namely, Q&A in session, and what they write in their opinions.)
Their scope of power is sufficiently limited that favor-trading isn't very important.
All that remains are the minor traditional formalities of when and where they meet and how they dress.
When your group grows beyond 9 people, that gets harder.
The internal politics are still interesting, though.
"The Brethren: Inside the Supreme Court" by Woodward and Armstrong looked (amongst other things) at why Nixon's new Chief Justice Warren Burger seemed unable to push the court in a more consistently conservative direction despite more conservative membership. Part of the conclusion was that he just wasn't that great at playing the internal political games required to stay in control.
Incidentally, the main source for that book was eventually revealed as one of the other Justices, Potter Stewart. It's a fascinating read.
Although Justices need not answer to anyone at all with perhaps the exception of defending themselves in their impeachment proceeding, the Chief does have some powers and duties, implicit ones such as designating authorship of a majority opinion when in the majority, and explicit ones such as administratively leading all of the Article II and III courts. The aclu's page is evenhanded and may be of interest. https://www.aclu.org/blog/speakeasy/role-chief-justice
Convention and precedent (and tradition) are also forms of hierarchical power. Those who assert them in the course of making a decision exercise power over those who intend to deny them.
