Thanks everyone. Jluetjen, the opposing cylinders clue cleared it up for me.
I did some calculations to try to figure out why the flat-six is "inherently balanced" but the flat-four is only in primary balance. What I came up with is that the accelerations of the pistons in the flat-four always sum to 0 and the same is true for the flat-six but when you look at the sum of the accelerations for the pistons on just one side of the motors, then the flat-six is much more balanced compared to the four. The calculations I did were for motors with stroke of 71mm and connecting rod lengths of 131 mm and these are the results I got.
Sum of piston accelerations for one side of flat-six.
Sum of piston accelerations for opposite side of flat-six.
Sum of piston accelerations for one side of flat-four.
Sum of piston accelerations for opposite side of flat-four.
For the flat-four, the maximum "unbalance" is 20 and occurs every 90 degrees of crankshaft rotation but for the six the maximum unbalance is only 0.01 and occurs every 30 degrees of crankshaft rotation.