Does the direction of travel matter for jet lag, when the time difference is 12 hours?

The direction you travel has no bearing on the severity of jetlag. Jetlag is caused by your body adjusting to the time difference. Direction only contributes to the length of time you are effected.

The difference between eastbound and westbound is more a matter of time of day you arrive. Eastbound long haul flights tend to leave late and arrive early. Combining tiredness from flying with the need to stay awake a long time usually results in day sleeping and prolonging the adaption.

West bound flights go with the sun, so when you arrive local night time is not far away, allowing you to get on local day / night cycle quicker.

Based in numerous east coast USA to SE Asia flights (basically 12 hours time difference) I prefer going west to arrive Bangkok at midnight and sleep. It is tough staying awake after arriving in the am on eastbound flights via Europe.

Ending up somewhere west of home (even if you flew east to get there) means that you need to stay up later in order to go to bed at an appropriate local time, and you may wake "too early" for where you are. Many people can stay up 3 or 4 hours late once without feeling too awful, and can happily "sleep in" the next morning. (And if it's a business trip, you can go to bed at 7pm local and no-one will even know.) Needing to stay up 8 or 10 hours late would not be perceived as easy.

If you end up east of home, you need to get up earlier, which most people find difficult, and a 3 or 4 hour shift may be very hard, never mind an 8 or 10 one. Again this has nothing to do with the travel process that got you there.

This summer a young adult of mine spent 24+ hours travelling to somewhere 12 hours different (and back) 4 times. All the options were covered - out west, home east, then out east, home west. The report I got was that it made no difference what direction the plane flew and even whether it was light or dark outside the plane or a transit airport. It was just very difficult. FWIW.

From personal experience, I find it easier to have a very long day (like 24 hours sunshine); I'll obviously be very tired, go to bed early, and wake up and everything is fine. The other direction, where 6 hours are cut out of a day and 6 hours cut out of a night, gets my timing totally confused.

You can do this at home.

Experiment 1: Sleep and wake at your normal time, but block your windows for several hours. Use something like cardboard which thoroughly blocks light. See how you feel. You have "compressed" your day.

Experiment 2: First compress your day by blocking off several hours of sunlight during the morning. Then try to go to bed and wake up several hours earlier, but block the sunlight during the evening, so that the period of sunlight is the same relative to your sleep/wake cycle.

Experiment 3: Full sunlight, but try to suddenly change your sleep/wake time. (I call it...daylight saving...) (This is actually probably the worst. Shift work where people routinely have to change their sleep cycle without actually changing location at all is notoriously bad.)

Thought experiment: Do you feel jet lag when traveling a good distance north during December or south during July? (Personal experience: No.)

Bonus thought experiment: Do people on flights following a polar flight path feel jet lag? E.g. Dubai <-> Seattle on Emirates.

Since this isn't blatantly obvious enough: Your friend is incorrect. It is the change in your sleep/wake cycle which causes difficulty, not the change in daylight. You will be completely miserable by changing your wake/sleep cycle without even changing your location (daylight saving), but you will be fine traveling north/south even though your period of daylight might change considerably (going on holiday somewhere in your same time zone).