My data is non-normal, I want to show my raw data, in a scientific journal, by median +/- mad, to show the true nature of the data.
However, if log-transformed, the data is normal. Can I then indicate significance based on calculation on log-transformed data? Which results in median+/-mad presented along with results from parametric tests.
My rule of thumb is that if you do a statistical test on a transformation of the data that you will plot, then you should plot that transformation of the data.
Ideally the transformation should be motivated by the data type; for example, suppose you are looking cell counts in a Petri dish. Since these grow exponentially (at least until they hit the limit allowed by the dish), a log transformation is well justified, both scientifically and in terms of making the data look more normal. In this case, the log transformed data better answers the scientific question of interest and eases the statistical methodology, so it's clearly the form of the data the reader should be interested in.
In the real world, sometimes log transformations are used almost entirely because doing so simplifies the analysis (e.g. taming outliers, etc). In this case, I would still plot the transformed data: this is the form of the data for which you are answering questions about (i.e. "are the means of the log-transformed data different?") and as such, the reader should be most interested in this form of the data.
Also, note that order statistics such as the median are preserved under monotone transformations such as logs, so the log of the median will be the same as the median of the log values - something that isn't true of the mean. For similar reasons, any distribution-free tests such as the Wilcoxon Rank Sum tests comparing two groups, etc., will give the same answer whether or not you perform the log transform.
If the log-transformed data appear to follow a normal distribution, then the original data follow a log-normal distribution. This implies that hypothesis tests that assume normality can be run on the log transformed data. Furthermore, the log-normal dsitribution is well characterized with the expression $\mathrm{GM}(X) \divideontimes \mathrm{GSD}(X)$, where $\mathrm{GM}(X)$ is the geometric mean and $\mathrm{GSD}(X)$ is the geometric standard deviation. Here, the symbol $\divideontimes$ denotes times or divided by and is used by analogy with the symbol $\pm$ that denotes plus or minus. The expression also tells you how to compute symmetric error bars.
