There is no such thing negative dimension in physics. At least I don't hear If there is any.
By the 1940s, topologists had developed a fairly thorough basic theory of topological spaces of positive
dimension. Motivated by computations, and to some extent aesthetics, many topologists began searching
for mathematical frameworks that extended our notion of space to allow for negative dimensions. It
wasn’t until the 1960s that one was constructed – the category of spectra. A spectrum is a generalization
of space that allows for negative dimensions. The study of spectra, called stable homotopy theory, is a
robust and elegant field. Go to here for more details.
Your analogy for understanding the dimension of space is useful for elementary cases. I mean at least till 3-dimensional cases, you can define the space by looking around, but in Mathematics or physics, The concept of higher dimension is also useful and Maybe because the negative dimension is not a useful concept, There never defined such a thing.
The concept of a higher dimension is very useful when you are dealing with a system of particles. When talking configuration space or phase space, we see how useful it is.
Just look at this simple example of dumbell,
In cartesian co-ordinate : $(x_1,y_1,x_2,y_2)$ with [ (x_1-x_2)^2+(y_1-y_2)^2=r^2 ]
In generalize co-ordinate $(X_{cm},Y_{cm},\theta)$ where $\theta$ is angle between x-axis and axis of dumbell.
The $(X_{cm},Y_{cm})$ can go anywhere but $\theta$ can take value from 0 to 2$\pi$. Thus if $(X_{cm},Y_{cm},\theta)$ denote 3D euclidean axis then C-space is region between plane $z=0$ and $z=2\pi$ ,If you remove the discontinuity in $\theta$ , you will find the C-space is actually hyper cylinder in 4D space.
