This is less an answer and more a request for further clarification from respondents, and maybe a clarification itself.
Here are a couple of examples from textbooks showing how the symmetry assumption for the signed-rank test is sometimes handled. On my bookshelf I couldn't find anything more clear or more explanatory.
There are also numerous examples from the internet and from Cross Validated where the symmetry assumption is stated simply and without further explication.
The Wilcoxon [signed-rank] test assumes that the sampled population is
symmetric (in which case the median and mean are identical and this
procedure becomes a hypothesis test about the mean as well as about
the median, but the one-sample t test is typically a more powerful
test about the mean).
-- Zar, Biostatitics, 5th, 2010, § 7.9
The important difference between the sign test and this [Wilcoxon
signed-rank] test is an additional assumption of symmetry of the
distribution of the differences.
- The distribution of each Di [difference in paired observations] is symmetric.
- The Dis are mutually independent.
- The Dis all have the same mean.
- The measurement scale of the Dis is at least interval.
-- Conover, Practical Nonparametric Statistics, 3rd, 1999, § 5.7
However, I think the responses by @jbowman and @Glen_b explain this assumption in a different light.
If the null is false, you don't necessarily require symmetry.
In some cases it may be convenient to assume a location shift alternative... However, that doesn't make it an assumption of the test, it makes it an additional assumption we've made for other reasons.
The Wilcoxon Signed Rank test does require that the paired differences come from a continuous symmetric distribution (under the null hypothesis, as Michael Chernick points out in comments.)
One way I might be able to sum up my understanding is as follows. The null hypothesis for the signed-rank test is that the differences are symmetrically distributed about a value. There are two ways this hypothesis could be wrong. 1) The differences could be symmetrical about a different value. 2) The differences could be not symmetrical. If #1 is the case, then the differences have a different location than in the null hypothesis, which is a useful result, and easy to interpret. If #2 is the case, then the data are skewed in a way that is a useful result. This may be a little less easy to interpret that in #1, but an examination of a histogram of the differences should reveal useful information about the distribution.
With this understanding, I think there is no assumption about the distribution of the original data, the differences, or the ranks of the differences.
How does this sound?