Loops don't get canceled out in the signature. (You might like to compute the signature of a circle. The second iterated integral is nonzero; in fact, by Green's theorem, it gives you the area inside the circle, maybe up to a factor of $\frac{1}{2}$ or something.)  What does get cancelled out is any segment where the path retraces or doubles back on itself.  The natural question is which paths have trivial signature.

The idea of signature was essentially introduced by K. T. Chen in 1958.  Chen has a notion of "irreducible" path, i.e. one which cannot be written as $\alpha \cdot \gamma \cdot \gamma^{-1} \cdot \beta$ where $\cdot$ is concatenation and $\gamma^{-1}$ is the reversal of $\gamma$.  Chen showed that for piecewise regular paths (piecewise $C^1$ with non-vanishing derivative), an irreducible path with trivial signature is trivial.  So any path with trivial signature is a finite sequence of "retracings".


> <cite authors="Chen, Kuo-Tsai">_Chen, Kuo-Tsai_, [**Integration of paths. A faithful representation of paths by non- commutative formal power series**](http://dx.doi.org/10.2307/1993193), Trans. Am. Math. Soc. 89, 395-407 (1959). [ZBL0097.25803](https://zbmath.org/?q=an:0097.25803).</cite>

More recently, in a very famous paper, Hambly and Lyons considered the case of a bounded variation path.  They introduce the notion of a "tree-like path", where in some sense every piece of the path is doubled back; but there might be infinitely many "pieces".  They prove that a path of bounded variation has trivial signature iff it is tree-like, and conjectured that this should still hold with less regularity assumptions.

> <cite authors="Hambly, Ben; Lyons, Terry">_Hambly, Ben; Lyons, Terry_, [**Uniqueness for the signature of a path of bounded variation and the reduced path group**](http://dx.doi.org/10.4007/annals.2010.171.109), Ann. Math. (2) 171, No. 1, 109-167 (2010). [ZBL1276.58012](https://zbmath.org/?q=an:1276.58012).</cite>

If you look up papers citing Hambly and Lyons, you can find more recent progress.  It appears that "trivial signature iff tree-like", appropriately defined, has been proved for weakly geometric paths.

> <cite authors="Boedihardjo, Horatio; Geng, Xi; Lyons, Terry; Yang, Danyu">_Boedihardjo, Horatio; Geng, Xi; Lyons, Terry; Yang, Danyu_, [**The signature of a rough path: uniqueness**](http://dx.doi.org/10.1016/j.aim.2016.02.011), Adv. Math. 293, 720-737 (2016). [ZBL1347.60094](https://zbmath.org/?q=an:1347.60094).</cite>

And for simple paths.

> <cite authors="Boedihardjo, Horatio; Ni, Hao; Qian, Zhongmin">_Boedihardjo, Horatio; Ni, Hao; Qian, Zhongmin_, [**Uniqueness of signature for simple curves**](http://dx.doi.org/10.1016/j.jfa.2014.06.006), J. Funct. Anal. 267, No. 6, 1778-1806 (2014). [ZBL1294.60063](https://zbmath.org/?q=an:1294.60063).</cite>

There has also been work on how to explicitly reconstruct a path, up to tree-like equivalence, from its signature.

> <cite authors="Geng, Xi">_Geng, Xi_, [**Reconstruction for the signature of a rough path**](http://dx.doi.org/10.1112/plms.12013),  [ZBL06775247](https://zbmath.org/?q=an:06775247).</cite>