Hmmm what seems to me from your statement;
"there will be two cells that will be almost opposites (for each chromosome, if one cell has the mother's chromosome, then the other will have the father's chromosome), except for the parts where crossing-over happened"
It seems to me; you're missing something.
It would happen if the haploid chromosome number n=1 and no crossingover is there.
If the haploid chromosome number is more than 1; i.e 2n is more than 2 (such as in pea n=7, 2n =14 and human, n=23 and 2n=46);
then at segregation event in meiosis-1 ; any-1 of 2 homologous chromosomes will go to a doughter cell; in a probabilistic manner (and this assortment is independent from other chromosome-pairs).
Take example. If all your somatic cells have a chromosome composition Aa Bb Cc (where with each letters I meant a chromosome and not gene);
then perhaps you are thinking it could be seggregated only into ABC + abc. But that is not the case.
In one cell at certain place at site of meiosis, it would be Abc + aBC; in some other cell it would be AbC + aBc; and somewhere else it would ABc + abC, and that way.
In addition; irrespective of how many pieces of chromosome you have; crossing-over will take place in probabilistic way (so; that will make different-gene-exchange at different cells at the site of meiosis.)
So ultimately you get not 2 but many more types of gametes.
Here is 2 simple drawings.
- Meiosis first division.
- Meiosis second division.
The book, Concepts of Genetics by Klug, Cummings and Spencer, 8th edition (the ebook I have), chapter 2 (mitosis and meiosis); clearly mentions the random-assortment (physical shuffle) event with examples.
here is a picture from that book.
diagram from Klug et al.
Reference: Concepts of Genetics ; Klug, Cummings and Spencer; 8th edition