Intuitively, what does the 1 mean in $\dfrac{a}{b} = \dfrac{1}{\frac{b}{a}}$?
For intuition, I instantiate variables with fruits: $a$ as apple and $b$ as berry (pick whatever berry you like).
In terms of apples and/or berries, what does the red $\color{red}{1}$ below mean? What's the unit of $\color{red}{1}$?
$\dfrac{a}{b} \quad = \quad\dfrac{\color{red}{1}}{\dfrac{b}{a}}$.
Unmistakably, I know $\dfrac{a}{b}$ means $\dfrac{\text{apple}}{\text{1 berry}}$, and $b/a$ means $\dfrac{\text{berry}}{\text{1 apple}}$.
I ask only for intuition. Please pretermit formal arguments and proofs. E.g. please don't rely on rationalizing the denominator .
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Your a and b are reversed in that equation because that equation corresponds to the tracks (0..1] and [1..) of this, that are symmetric.
The 1 / x plot is [at least] doubly symmetric. One of those two symmetries is y = x. That one's the symmetry axis you are asking about.
HTH.
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I think applying intuition based on dimensional analysis here will just serve to confuse rather than illuminate.
$$\frac{1}{\frac{\text{berry}}{\text{apple}}}$$
would simply read as 1 over berry per apple but that itself does not have much intuitive meaning. It means that you are dividing 'dimensionless' 1 by berries per apples even though this is in the end equivalent to apples per berry ($\frac{\text{apple}}{\text{berry}})$. This is simply not an useful representation of a relationship.
Simply put, not all representations of mathematical expressions are equally useful. Consider the following example:
$$y= 2x +10 \Leftrightarrow y-x -5 = x + 5 $$
In the equation of line on the left we can clearly interpret 2 as the slope of the equation and 10 as an intercept of the equation.
In the equation on the right that is completely equivalent to the one on the left we cannot directly interpret any of the numbers.
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