We know what you're thinking and the answer is yes.
These monuments do not represent The Inverted Circle. Some of these monuments were never raised, never conceived of. These read "You are here". Those that were in memory of events that may yet happen, say only "You are not". Ignore them. Ignore them all. They signify nothing. They do not signify nothingness. There is no thing or not-thing which they represent. There is an absence of signification.
The facts of the membership of The Inverted Circle are so secret, so utterly hidden from knowledge that you or I may have belonged, may have founded it, yet we would not know. It is said that to not know of The Inverted Circle is to be a member, and to hear of it – to believe it exists – is to be expelled, expunged from the ledgers. It does not exist for those who believe in it; it exists only for those who do not. Having read this, you must by necessity believe both its existence and non-existence are undeniable.
What could be the goals, what the purposes of such a society? Some believe the aim is to remain hidden, others that it plots only to hide this plot from its membership. Must it have goals at all? We cannot say.
There is an infinite multiplicity of such societies, all of which have been founded at every hypothetical point in the history of reality, constantly and concurrently being founded and dissolved or continued arbitrarily – but no Möbius strip they. The Möbius strip represents a loop, an infinite repetition of returning to the point of origin. The Inverted Circle instead is a one dimensional line in two dimensional space with a constant curvature: to our eyes it appears to repeat yet as it possesses no width it can never collide with, never overlap itself. It is a compound spring, infinitely long and wound infinitely tightly.
What will come of this? All that can be known for certain is that The Inverted Circle and all hidden societies like it will continue, as they always have. They will always be with us.
These monuments represent nothing.