Is the thingie/cothingie distinction absolute? 
Is there some inherent quality of a mathematical object that marks it as being "naturally" a thingie or a cothingie?

Suppose, for example, that two mathematical concepts, say, doodad and doohickey, originally defined far, far away from the reaches of category theory, are later discovered to be–who knew?–categorical duals of each other.

Would it then be entirely arbitrary whether to rename doohickeys as "codoodads" or rename doodads as "codoohickeys"?

The fact that, for me at least, cothingies are typically far more exotic than their duals suggests that there is no intrinsic "coquality" that distinguishes cothingies from thingies, and therefore, all other things being equal, it is the more exotic member of the pair that gets the coname.  (Otherwise the extreme underrepresentation of cothingies in my prior mathematical education is hard to explain.)  But occasionally I come across comments that suggest that some mathematicians at least have an intuitive sense of how to classify arbitrary mathematical entities as either thingies or cothingies.  Therein the nundrum.
Thanks!
PS: Wikipedia's extremely useful List of mathematical jargon desperately needs a knowledgeable entry on the term cointuition.
 A: In category theory, cartesian squares, products, kernels, and limits are all maps into a given diagram, while cocartesian squares, coproducts, cokernels and colimits are all maps from a given diagram. Homology and cohomology follow this rule as well (when taking resolutions for covariant functors).
As Henning pointed out, this shouldn't be confused with co-vs-contra.
To answer the title of the post, I don't believe "co-ness" is absolute, but the above seems to be a pretty good guiding principle. Sometimes it may just be a matter of which is discovered first.
A: I'll just give one example showing that the distinction is indeed not absolute.  
Traditionally in differential geometry you define tangent vectors to a manifold $V$ at a point $P$ first, for example as the vector space $T_P(V)$ of equivalence classes of differentiable curves through $P$ .
And then you define the cotangent space $T^*_P(V)$ as its dual vector space.  
Zariski realized that in algebraic geometry, if you consider a variety $X$ and a point $x\in X$, it was more natural to first define the cotangent space of $X$ at $x$ as $\Omega^1_x(X)=\mathfrak m_x/\mathfrak m_x^2$.
(In this formula $m_x$ denotes the maximal ideal of the local ring $\mathcal O_{X,x}$ of $X$ at $x$)
The tangent space is then defined as the dual of the cotangent space $\Omega^1_x(X)$, seen as a vector space over the field $\mathcal O_{X,x}/\mathfrak m_x$.
This approach (published by Zariski in 1947) has technical advantages, especially  in the case when $x$ is a singular point of $X$ or for varieties in characteristic $p\gt 0$. 
A: An algebraic category like $\text{Grp}$ or $\text{Ring}$ often has a forgetful functor to $\text{Set}$ which has a left adjoint (the free object functor) but generally not a right adjoint. It follows that the forgetful functor preserves limits but generally not colimits. That's one reason you might consider limits more basic than colimits, at least if you like algebra, since limits look more familiar from $\text{Set}$ (e.g. the underlying set of the product is the Cartesian product in $\text{Grp}$ whereas the underlying set of the coproduct is very different from the disjoint union). 
On the other hand, arguably the status of the Yoneda embedding $C \to \text{Set}^{C^{op}}$ as the free cocompletion of a category is evidence suggesting that colimits are more fundamental. 
