I've heard and would often use "flipped around" for this:

"I flipped the box around."

To me, that means you pivoted it 180 degrees, unless more context is given. It could mean you turned it upside down, but only if that's obvious from the context. If you want to say you turned the box upside down, without any other context, you would say:

"I flipped the box upside down."

Also, just "turned around" works. I would never say "leftside right".

No. I'm a native British English speaker and wouldn't have a clue what you meant by "leftside right".

I would suggest describing the action in terms of positioning the faces (left, back, right, front, top, bottom), e.g. "Rotate the box so the left side is facing the front".

"180 to the left" is the most natural way I can think of to express it.

In skating/snowboarding, it would be a "backside 180", assuming regular stance ;)

I think "back to front" is what you are looking for, but in my experience as a Canadian, it is not a very common term in American English.

The synonymous term in American English is "backwards". That said, the connotation might be a bit different.

For example, using it the way you used "left-to-right" it may not be a perfect fit. "I turned the box backwards" has some implication that the original position was where it was supposed to be, but I believe most would understand it.

You can say whatever you want in English.

But unless you're trying to be funny/clever/etc., yes, I would refrain from saying "leftside right", or "rightside left" for that matter.

"Leftside right" is likely unclear to many native speakers just because it's a word and phrase that isn't used.

Inside-out, Upside-down, and other such phrases are in the vernacular, as are the words "inside" and "upside" (as well as "outside" and "downside").

"Leftside" and "Rightside", however, are... less common.

Alternatives

Around is a fair substitute: "I turned the box around."

See also:

"She turned around and looked."

"He really turned his life around."

"If you kids don't stop, I'm going to turn this car around!"

The most common way of telling someone to turn something so that it is facing the other direction is:

turn around.

This is the usual way to say 'turn something 180 degrees from side to side'.

If it is facing to the left, then afterwards it should be facing to the right.

If it is facing you, then it should be showing their back to you afterwards. This is often misinterpreted as turning a complete circle so as to face the same way again afterwards, but that is hardly ever what is intended.

You can say 'leftside right' and that would be understood, but it is not a thing any body actually says ever, at least not for telling someone to turn something. Urban Dictionary has a description meaning reverse like in a mirror, but I can't find any other web instances of it actually in the wild.

In most cases where one rotates an object 180 degrees about a horizontal axis, it won't matter which horizontal axis one is using. What will matter is that the parts of the object that were oriented up are now oriented down, and vice versa. It wouldn't generally matter whether the parts that had been in front stay in front, or the parts that had been on the left stay on the left, or neither, so there's no brief term that makes such distinctions.

Rotating a box 180 degrees around a vertical axis will flip it left-to-right and front-to-back. Such a move is generally called "turning it around" or perhaps "flipping it around".

For operations which flip an object around a single axis only (typically only applicable to operations on a computer, images, or other "virtual" objects), one would typically say the object was flipped top-to-bottom (not upside-down!), left-to-right, or front-to-back. Note that "flipping an object upside-down" typically involves flipping it about planes that are perpendicular to a vertical axis and another unspecified axis, and "flipping the object around" would involve flipping it about planes perpendicular to both front-to-back and left-to-right axes, while flipping top-to-bottom, left-to-right, or front-to-back would only involve reflection about a single plane.

tl;dr: be clear.

As Hot Licks points out, you could -- the trouble is, people wouldn't understand you.

If you need your listener to know how you turned the thing (or which way up it is now), you will need to make that clear -- and your suggested "leftside right" fails to make it clear. Even if your listener understood that the side that used to be on the left is now on the right and vice versa, you haven't said whether you turned it around a vertical axis or about a front-back axis. What is the best thing to say? I think it depends to some extent on what is plausible in your context (is the object so heavy that it can't be turned about a horizontal axis even though it can be slid and turned around a vertical axis?).

Anyway if the context doesn't resolve the ambiguity, I'd favour being explicit even if that means a lot of words, e.g. "I turned it upside-down so that its left side is now on the right" or "I turned it around so that its left side is now on the right". Just "I turned it around" (as Mitch suggested) sounds nice but I still have a nagging suspicion that in some contexts people might not be sure which way you turned it.

Other answers like "flip the box around" sound natural, but if you're looking to maintain the focus on where the left side ended up, I would go with "I turned the left side of the box towards the right."

As a native English speaker (I'm English), I'd just say "turn it 180" - the direction doesn't matter unless it's important.