Upside-down equation: algebra puzzle

This quirky equation was presented to me by a fellow teacher, along with the instruction "solve it both ways up".

The equation is simple enough to solve, with one integer root. What's more interesting is that the page can be rotated $180$° and a new equation can be read.

This second equation has two roots. I found it fairly satisfying that both of its roots are integers, but I was hoping that it would share a root with the first equation.

I set about looking for an equation that makes sense when read upside-down, where the two equations share a root.

My students immediately came up with trivial examples like "$x=1+1$" and "$1=\frac{1}{x}$". I haven't found anything more interesting yet. I haven't even figured out an approach better than trial and error, with different permutations of the digits $1$, $6$, $8$ and $9$.

Can anybody help me to find such an equation, ideally one that is similarly "difficult"/"interesting" as the equation presented to me?

Edit: credit is due to Rob Eastaway who (I have learned since posting this question) originally posted the first image on Twitter.

There's of course

$$ \frac{1 + x}{1} = \frac{1}{x} ~~~\mbox{and}~~~ \frac{x}{1} = \frac{1}{x + 1} $$

These two don't have any integer roots, but they have the golden ratio as one

If I may be allowed to use $2$ and $5$ as well (their digital representations can be flipped ... and result in themselves):

$$\frac{82-x}{59+61-x+51}=\frac{x-28}{65+x-19+15}$$

Both original and flipped version have $x=55$ as a solution

Here is a digitized version:

and its flipped version: