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Why we can analytically define ε in dual numbers so to distinguish ε from −ε but cannot do so in complex and split-complex numbers?

Why we can analytically augment the algebraic definition of $ε$ in dual numbers so to distinguish $ε$ from $-ε$ but cannot do so in complex and split-complex numbers? Is there a theorem on this?

Algebraically one cannot distinguish between $\varepsilon$ and $-\varepsilon$. But it is possible to augment their definition analytically so to distinguish them.

In dual numbers there is a common equality: for differentiable at $x=a$ function $f(x)$, $f(a+b\varepsilon)=f(a)+b\varepsilon f'(a)$.

Now, one can define that if at point $x=a$ $f(x)$ has right and left derivatives $f'_r(a)$ and $f'_l(a)$, and they are not equal, then
$f(a+\varepsilon)=f(a)+\varepsilon f'_r(a)$ and $f(a-\varepsilon)=f(a)-\varepsilon f'_l(a)$. In other words, $\varepsilon$ is defined as a positive infinitesimal, and $-\varepsilon$ is defined as a negative infinitesimal. This provides an optional analytic definition distinguishing $\varepsilon$ from $-\varepsilon$, but the algebraic structure can work just well without such additional analytic property (but with it one can evaluate more functions at more dual numbers).

Still, in dual numbers one cannot distinguish $\varepsilon$ from $a \varepsilon$ when $a>0$ even with this analytic addition.

So, my question is: why is it possible to define dual numbers in such a way so to distinguish the sign of dual unity, but the same cannot (?) be done with sign of imaginary and hyperbolic unities and with scale in dual numbers?

Is there any strong argument, why?

Is it because the lexicographical ordering in dual numbers naturally embeds into ordering of reals, while in complex and split-complex numbers it does not?

[The same question on Mathoverflow](https://mathoverflow.net/a/423799/10059)