Until more hints are provided, the only thing I can really supply is a dump of the backwards and forwards versions of all of the cyclic rotations of all of the coprime sampling intervals of the ciphertext. That is 2*123*80 = 19860 lines. I hope that is useful to someone. I wrote the code in Haskell. The file is 2MB, so I am splitting it up into several parts to fit it all on pastebin.

Part 1: http://pastebin.com/mD2DuL5c

Part 2: http://pastebin.com/BPCNWwa3

Part 3: http://pastebin.com/XW61r97s

Part 4: http://pastebin.com/5u0DPewa

Part 5: http://pastebin.com/6wBX96EE

EDIT

I have made no changes to the dumps above, but I just thought a bit more about the nature of this problem and came to a number of impressions/conclusions, which are not exactly rigorous but have some basis.

From the information provided, both in the post and in its comments, I have arrived at several conclusions:

Automorphism

Because the decryption does not rely on the orientation of the cyclical ciphertext or the starting point, we can consider this problem from a modular-arithmetic standpoint. We also know that all of the characters in the plaintext $p$ are preserved in the ciphertext $c$, merely subjected to a rearrangement. We can therefore treat the cipher as an automorphism, because it is a mapping of $\mathbb{N}_m \rightarrow \mathbb{N}_m$, where $m$ is the length of the message. We will therefore consider $E$ to be the encryption function, and $D$ to be the decryption function, where $D(E(p)) = p$. We also know that, for an arbitrary cyclic shift $C$, we have $E(C(p)) = C(E(p))$ and $D(C(c)) = C(D(c))$. We thus conclude that $E$ and $D$ are pure, stateless functions, meaning that they produce the same result for the same output independent of any history. We also know that $E$ and $D$ depend only on the positions of letters, and not on their values. For,if this were the case, the size of the domain of $E$ would be $m*L$, where the alphabet of the messages is of size $L$, while the range would still be $m$, as the letters themselves are merely transposed and not transformed; by the Pigeon-Hole principle, there must therefore be multiple index-value pairs that map to the same index of the ciphertext, which we know not to be the case. We therefore consider $e$ instead of $E$, which is the function mapping an index in plaintext to the ciphertext index when $E$ is applied, which we state as $$p_i = c_{e(i)}\;\forall\;1 \le i \le m$$

We therefore only need to determine the mapping of $e$, which we know must be linear if $E(C(P)) = C(E(P)) \implies e(i+x) = e(i)+x$. After I create a table of the positions of each letter in ciphertext and plaintext, I will post the results of that.