I believe the quote I was thinking of was the ‘John Galt Pledge’ from Part Three, Chapter I of Atlas Shrugged:
I swear by my life and my love of it that I will never live for the sake of another man, nor ask another man to live for mine.
This fits the criteria of ‘renouncing control over another in exchange for them doing the same’ and of being a ‘political or philosophical work, perhaps an anarchist or libertarian text’, but lacks (a) the multiple examples and (b) the specific mention of a whip that I stated in my original question.
However, from the exact wording of having the ‘whip hand over’ another in my original question my best guess is that I may have been conflating the quote I was looking for with UK MP Enoch Powell's completely unrelated and rightly infamous 1968 “Rivers of Blood” speech, which includes with the following line:
In this country in 15 or 20 years' time the black man will have the whip hand over the white man
Another potential knock against this answer is that I have never read Atlas Shrugged, however I feel that the ‘John Galt pledge’ is exactly the kind of phrase that one would be likely to run into independent of the original source (e.g., here), and so this is less salient of a point.
In the absence of any other more compelling answers in the two and a half years since I asked the question, I am assuming that the Atlas Shrugged quote is the answer I was seeking, and that the mention of whips was a misremembered red herring.