I am no genius but the crossover is only or mostly to simply put the wire in the opposed position for layout purposes. A regular wind would have your wires both exiting at 6 o’clock. The crossover gets you in at 6 and out at 12. Edit to add the theoretical benefit of the crossover is to put distance between your just cleaned up line away from your “dirty line”.
Here are wrapping instructions for transformers for when you get to that point in your toroid usage. It is a great read for transformer type usage but not for choke usage like you need at the moment but do throw it in your saved radio reading files or bookmarks.
http://www.gnarc.org/wp-content/uploads/The-End-Fed-Half-Wave-Antenna.pdf
I am going to look for another site link, I think Palomar has a page on that if I can find it.
I didn’t see what I was looking for on their site, I may be remembering a youtube video they had out.
here is a link to a picture of a kit they sell for the ubiquitous IC7300 and it shows the common mode chokes they used. All 31 material ferrites iirc.
https://palomar-engineers.com/wp-content/uploads/Palomar-Engineers-Icom-7300-RFI-Kit-Flyer.pdf
5-7 wraps through a toroid for the coax and power supply I noted.
Here is one of their presentations and similar to what I recall watching.
https://palomar-engineers.com/wp-content/uploads/ABCs-of-RFI-for-Hams-SPARC-10282021.pdf
and since it was handy I popped the cover off my DXE max core 1:1 Balun. Inside are 2 cores that are most probably 240-31. Coax wrapped through center eleven times, 5, the crossover, and 5 more. There is a blue unmarked capacitor across the coax on the post side of the car as it is built to be a center feed choke for a dipole. I am guessing it is a high voltage 100pF rated cap. While it is a dipole box,....its main purpose is to choke common transmit power from radiating back on the coax to avoid making your dipole into a triplole.
Not loose individual wire though, sorry.