Designing a non-inverting programmable gain amplifier circuit using the LT6910-1
First of all, ask yourself if you really need a non-inverting amplifier. You suggest that you are processing &quot;sound&quot;, in which case it might not matter whether the signal is inverted or not. Maybe you are doing something else with your signal which requires it to be in phase with the input, but if you are just going to connect the output to a speaker then it doesn't matter.
Assuming you can't just use the PGA as-is with its signal inversion, there is no way to use that particular PGA both (a) as it is designed and (b) as a non-inverting amplifier -- the datasheet for the PGA says in page 1 that (emphasis added):

The LTC6910-Xs are inverting amplifiers with rail-to-rail output.

The datasheet also includes a block diagram which shows how the amplifier is configured internally:

Looking at the block diagram you could theoretically abuse the device by swapping the use of the IN and AGND pins since they are, respectively, the inverting and non-inverting inputs of the internal amplifier. This isn't a great idea, though, since the device was not intended to work this way. Also, AGND is not rail-to-rail input (unlike IN) and the minimum gain would be a gain of 2 (which would be a problem if you require a gain of 1 to be a possibility). You can try it if you absolutely have to use this particular PGA, but I don't recommend this approach.
A better solution would be to just use a non-inverting PGA. You can search for a suitable PGA across multiple manufacturers at places like Digikey (this search will get you started, and you can filter based on additional needs such as rail-to-rail capability).
Finally, you can add another inverting amplifier stage before your current PGA circuit so that the double inversion results in a non-inverted output. You can either use a dual amplifier PGA (it looks like the LTC6911-1 is a dual amplifier equivalent of your LTC6910-1) or a simple op amp configured as an inverting amplifier. If your minimum gain selection is greater than 1 you could possibly improve performance of your overall amplifier by configuring this extra amplifier with some gain in order to split the total gain between the two stages.