One of the best things about 23andme is its transparency!! In addition to its awesome api that allows users to build tools to analyze their genetic data, the 23andme website also contains an exploratory mode which allows users to search for specific single nucleotide polymorphisms (SNPs). SNPs associated with 23andme health reports in addition to other traits are available on SNPedia. Some of these traits include SNPs for left handedness or even SNPs for empathy. To get this information simply search for a SNP or trait of interest on SNPedia such as empathy: Rs53576. Then login to your 23andme and go to https://www.23andme.com/you/explorer/.
Here you can search for any SNP of interest like Rs53576. My genotype at this SNP is GG.
If we go back to SNPedia in the left hand corner we can see that having GG at the position indicates I may have my optimistic and empathetic tendencies encoded in my genome :)
Some of these SNPs arnt as well studied like the SNP for left handedness rs1446109-rs1007371-rs723524 here if you have something like AA-TT-GG you are likely to be left handed. Also at the top of the image above, if the orientation does not say "plus" but instead "minus" you need to use the reverse complement of the 23andme genotype to determine your genotype at that SNP.
Happy Hacking :)