The advantage of asymmetric objects We know that it is usually much easier to work with highly symmetry objects, the objects that have many automorphisms like the sphere, Lie groups, complete graph,... But is there any advantage of asymmetric objects, the objects that have only trivial automorphisms and what is that advantage? For example, is there a category $C$ and property $P$ that true for all objects of $C$ that it's easier to prove asymmetric objects have property $P$ in that category than for symmetric ones?
Motivation: Give a category $C$, we want to prove all objects of $C$ have property $P$. I have an interesting idea as follow:
For each objects $A$, let $G_A$ be a automorphisms group of $A$. Assume for each group morphism $H\rightarrow G_A$, we have a nice quotient $A/H$, and for some group $H$, it's easy to prove this statement "if $A/H$ has property $P$ then $A$ also has property $P$". For example, we can prove that statement for all cyclic groups, because for each non-trivial group $G$, there exists a non-trivial cyclic group as its subgroup. So we keep quotienting objects $A$, and if we are lucky, we will reach to an asymmetric object in $C$, and now we just need to prove that all asymmetric objects have property $P$.
 A: My favourite example is related to some bit of work of my own, so I apologise in advance for a bit of self-promotion. It concerns dealing with symmetric operads (algebraic ones, meaning that the $n$-th component is a representation of $S_n$).
Presence of symmetries, while good in theory, complicates things in practice: it is harder to choose canonical bases that are convenient for calculations etc. The solution is to... forget about symmetries. If one does that naively, the result is very disappointing: for example, if one takes the operad of Lie algebras and considers it as a nonsymmetric operad, it becomes a free operad on infinitely many generators (it is not possible to even describe the Jacobi identity in the realm of nonsymmetric operads, so the dependencies disappear), as proved by Salvatore and Tauraso some 15 years ago. This happens because by considering nonsymmetric operads, we forget some part of the story alongside with the symmetries, some of the structure maps.
How to forget about symmetries in a smart way? The solution is to consider "shuffle operads" (introduced by myself and Khoroshkin): symmetric operads are algebras over the monad of trees, shuffle operads are algebras over the monad of shuffle trees, which just choose a particular way to draw a tree on a plane, so no structural information will be lost. This does exactly what you are talking about: eliminate automorphisms, and the gain is quite noticeable (we did it to define Gröbner bases for operads, and that helped a lot in quite a few questions of algebra and homotopy theory). See our article for more details.
A: Such objects are generally known (in the fields I’m familiar with) as rigid, and yes, working with them can often be very useful.
The advantages I’m familiar with come from the fact that if objects of some type — widgets, say — are rigid, then essentially nothing is lost by working with “widgets up to isomorphism”. A typical example is well-orderings.  These are always rigid; their theory is traditionally developed mainly in terms of ordinals, which are canonical representatives of each isomorphism class.  The rigidity ensures that every well-ordering is uniquely isomorphic to some ordinal, and that’s simplifies several aspects of the development of the theory.
By contrast, consider some non-rigid objects — say, finite Abelian groups.  Even if we have some way choosing of representatives of isomorphism classes (e.g. by the classification theorem), it’s not true that every finite Abelian group is uniquely isomorphic to its chosen representative.  In other words, finite Abelian groups can’t be naturally identified with their chosen isomorphic representatives.  It turns out that this means working with chosen representatives isn’t nearly so fruitful as in the rigid case.
A way of saying all this in fancy languages is that families of arbitrary objects typically form a (pre-)stack, but families of rigid objects form a (pre-)sheaf — a technically much simpler notion.
The nlab has a page on rigid objects mentioning several more examples.
A: In light of your stated motivation, the following may not be what you had in mind, but objects with no symmetries are often easier to handle when it comes to computation and/or enumeration.  For example:

*

*Determining whether two (finite) graphs are isomorphic is usually easy when the graphs have no automorphisms.


*There is a nice formula ($n^{n-2}$) for the number of vertex-labeled trees with $n$ vertices, but if the vertices are unlabeled, then you have to account for the automorphisms, and the best you can do is some messy "formula" based on the Redfield–Pólya theorem.


*If you're trying to use a SAT solver to prove that an instance of the Boolean satisfiability problem is unsatisfiable, it is usually helpful to add constraints to break symmetries.  Otherwise, the solver may end up doing essentially the same (futile) search over and over again before finally concluding that no solutions exist.
