I think the next number must be
That would be because each number in the series
consists of two parts that describe the previous number: its length, and an inventory of its digits. These parts are written one after another, but their order alternates, as suggested by the story. (Reversed order when in an 'odd' country.)
The format of the inventory is "for digits 0-9 in order, count how many times each digit occurs, and write those numbers together, omitting any leading or trailing zeroes"
So for example
to get from 102132413 to 913221, notice that 102132413 has 9 digits, out of which 1 is zero, 3 are ones, 2 are twos, 2 are threes and 1 is a four. (five to nine omitted, since there were none.)
For the next step,
to get from 913221 to 2210000016, now the inventory comes before the length: no zeroes (omitted), 2 ones, 2 twos, 1 three, 0 fours, 0 fives, 0 sixes, 0 sevens, 0 eights and 1 nine, followed by the length, which is 6.
The name of the sequence (and the pun in the title) might refer to the fact that
you can also get the length of the previous number by summing up the digits in the inventory part, and the capital sigma letters ($\Sigma$) in the name are typically used to denote a sum. Therefore, the two parts might conceivably be called the "sum" (or even "awe-sum"), and the "sequence".