Any idea what causes different harmonics based on the direction of strum/vibration of steel string ? Up / Down

The pitch is the same but the sound varies slightly. I guess it has something to do with angular momentum but that is guess.

I think this is entirely down to the mechanics is your fingers/pick. If you place a pick perfectly orthogonal to the fret board and mechanically pull it either direction you would not hear a difference. But when you are picking downwards the pick flexes up more than it flexes down on an up-pick, just based on how you are holding the pick.

It definitely has nothing to do with which direction the string vibrates, because it's harmonic oscillaton. It vibrates just as much both directions, no matter which direction it starts in.

The string itself has symmetry but the boundary conditions at the ends are different in different directions. They’re not perfectly rigid, so there’s a little energy exchange between the string and other things vibrating on the guitar, including adjacent strings. This coupling to other stuff probably explains the differences you hear.

This was my thought as well. The strings of a guitar rest on little grooves at then ends (bridge at the bottom, nut on the neck), so there is a force affecting from three sides of the string (sides and underside). Plucking up makes the string "jump" slightly on the slots, so there is less force on the upward phase of the oscillation and more on the downward phase. Plucking sideways will not have a difference between the forces.

Even plucking sideways would cause a close to circular movement in the string.

I think that's only the case for theoretical strings. A real one on a guitar made out of existing materials would have different properties in different directions, even if the design is perfectly symmetrical wood is asymmetrically anisotropic.

Strings also rarely vibrate in 1 dimension. After a few oscillations it turns into a much more complex movement.

I don't think either of those effects matter when comparing an idealized "up or down" stroke. As in, it's not the direction but the mechanical attack that does it. If you turned the guitar upside down it would sound the same (on the same string)

Inherent frequencies of the structure holding the string have a lot of effect on the sound, that's why you can hear the difference between a western style guitar, a semi-hollowbody and something like a sitar or oud.

I'm not sure why turning the guitar around would matter in either scenario, unless you're planning to mirror it exactly. And if you could do that, the sounds would be flipped as well. At a fundamental level a guitar is not symmetrical, and the sound would change even if you were able to perfectly reverse your picking technique.

Find a stroboscope and play around with it while jamming. If the frequencies line up you can see the string movement change over time, it's really cool but much more complicated than a 1D approximation.

We are talking about what causes the difference between strumming a string down vs up. If you hold your guitar upside down, pointing right, and strum, I would expect you to hear the same qualitative difference. Meaning it's not about the direction of the strum as much as differences in the mechanics of the pick and strumming hand.

Yeah, but answering OP's question with "its because you strum different" is pretty obtuse lol

Obtuse? It's the truth. OP was asking if there is a difference between the string swinging up or down, and I'm explaining that no, there sort of can't be. Plus suggesting an experiment to demonstrate this.

I think there are lots of factors. The distinction between tone is due to the initial transients, if you take a daw and chop out the transient and only compared the sustained tail of the notes, one would be hard pressed to distinguish an up from a down stroke on a single string.

The transients are more interesting and relevant for this question though. I think bio-mechanics of the hand aside, the main difference between an up and down stroke on a single string is the difference in the pick angle. Usually one has a slight downward angle on a down stroke, i.e. pushing the string towards the body of the guitar, versus an up stroke would lift the string slightly away from the guitar. The interaction of the string with the bridge and nut will be different in these two scenarios, creating different transients.