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First, a little notation. Call a the first chromosome from my mother and A its homologous partner from my father. b and B would be the second pair, c and C the third, and so on, down to w and W (the 23rd).

The tetrads in interphase would be expressed as aa AA...ww WW (so, for example, AA are sister chromatids). Finally, use the notation a' and A' to express that those two chromatids have crossed over via recombination.

Ok: I understand that in Metaphase I, there's a random assortment of chromosomes at the equator, so that we might end up with a division like so:

aA'    Aa'
bB'    b'B
CC'    c'C
...
wW'    Ww'

Which pair ends up on the right or left is random, and there are therefore $2^{23}$ possible arrangements.

My question is whether an analogous shuffle happens during metaphase II? I visualize metaphase II as taking one of the columns above and turning it on its side, so that for instance taking the right column you have:

a' B  C  ... w'
A  b' c'     W

These "rows" are then split to become 2 of the 4 gametes.

The question is whether at this stage, before this final split, there is an analogous shuffling possible whereby, say, the A and a' could switch sides?

My intuition is yes, of course, but in all the accounts of meiosis I can find online, the "random assortment" and shuffling is all described as happening in metaphase I.

(I think whether the shuffling between this particular A and a' happens in metaphase I or metaphase II is inconsequential in terms of which one ends up in which daughter cell... but I'd still like to have a sense for what's literally physically happening.)

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Metaphase I

Your understanding of the metaphase I is not entirely correct. In normal (human) metaphase I homologous chromosomes separate. Therefore, sister chromatides go together. More precisely sister chromatides are kept together at centromere. You have only two possibilities for each chromosome:

<-AA   aa->

or

<-aa   AA->

If we multiply the two possibilities for all 23 chromosome pairs it leads to correct number 223 of possibilities.

There can be one or more crossovers between two or more chromatides but the fate of crossovered chromosome arms is determined by the fate of centromere they get connected to.

Your diagram of random assortment, if I understand it correctly, allows for more than 2 possibilities for chromosome pair:

<-Aa   Aa->
<-aa   AA->
<-AA   aa->
...

And it would lead to incorrect number of possibilities (more than 2)23

Metaphase II

In metaphase II sister chromatides separate as they would in mitose. Yes, there is random asortment of which chromatide goes left and which goes right. With your notation it could look like this:

<- A   A'->
<- b   b'->
<- C'  C ->
<- D   D'->
...
<- w'  w ->

However, as I said earlier, some chromosomes can crossover more than once, so you can have thinks like:

<- g''   g'->

or

<- F'''   F ->

or

<- h'    h'->

Inverse meiosis

For every rule in biology there is an exception. Some species have inverse meiosis. This means they do it in reverse order. In metaphase I they separate chromatides and in metaphase II they separate homologous chromosomes.

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