Is it valid to average correlation coefficients in this scenario? We have a client who manufactures materials with different surface 'roughness' characteristics, ranging from near-glass smooth to coarse sandpaper-rough.
They have ten different grades of roughness, from 1 (smooth) to 10 (coarse), and their engineers are trained to be able to identify each specific grade by touch.
Engineers undergo periodic training sessions, in which they have to identify the roughness grade of 10 random samples.  Each engineer's responses are entered into our software, which produces a Pearson Correlation Coefficient as a measure of each engineer's performance in each test session.  We supply the client with these results - i.e. a table resembling the following:

Questions:


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*Is it valid for the client to calculate the mean of the values for (e.g.) Session 1, in order to say that the average score for all engineers in Session 1 was (0.98 + 0.87 + 1.00) / 3 = 0.95?

*Likewise, is it valid for the client to calculate the mean for each engineer, in order to say that (e.g.) the average score for Helen Jones across the two sessions was (1.00 + 0.88) / 2 = 0.94?
Our client is not planning to use these numbers for any serious statistical analysis, but rather just to get a feel for whether the training sessions are helping the engineers to get better at judging the roughness of the materials.
Edit 1
Only integers are allowed for the engineer's responses. The correct answers correspond to defined roughness standards measured on a scale of 1 to 10.
I've looked again at the client's original spec. They are not looking to determine if the engineer can identify a specific roughness number for each sample (my mistake for stating this in my question). Rather, if the engineer can consistently distinguish rougher samples from smoother samples, then this should lead to a higher score.  
So, if in a particular session the 'correct' answers are 1, 2, 3, 4, and John answers 1, 2, 3, 4 and Helen answers 2, 4, 6, 8, the fact that they both result in a correlation coefficient of 1 is fine for the client - they have both demonstrated that they score higher values for rougher samples.
 A: The correlations appear to be measuring the wrong quantity in the first place, so the question may be asking about how to summarize something that doesn't and can't perform as desired. 
As I understand it, the correlations are measured for each engineer's reports as one variable and some independently obtained value for the true or correct roughness for the other variable. I don't gather from the question whether non-integers are allowed for either variable. 
But a strong correlation could arise if an engineer were biased but otherwise reliable, e.g. if John's grades are given exactly by true $-\ 0.5$ or Helen's by true $+\ 0.5$ then their correlations would be $+1$. Conversely, any engineer for who there was deviation from engineer's $=$ true could not score a perfect correlation, even if they were on average unbiased. Otherwise put, correlation measures linearity of relationship, not agreement of values. The fallacy is seen most starkly by realising that $y$ and $by$ have perfect correlation for any positive value of $b$. 
This may not bite very hard for your data, but if you desire a measure of agreement, then correlation can at best only be such a measure by accident. Concordance correlation is such a measure, although over-reliance on any single-valued reduction of any data is likely to miss important structure. 
The forum contains various other threads mentioning concordance correlation, e.g. Does the concordance correlation coefficient make linearity or monotone assumptions?
In general, averaging correlations makes most sense when they encapsulate all the information of interest. Here that seems unlikely at best. 
