Question: It's common for students to be introduced to tensors as "things that transform like tensors" - that is, their components must transform in a certain way when we change coordinates. However, we can do better by defining a tensor as a multilinear map from $ V\times...\times V\times V^\ast\times ...\times V^\ast\to \mathbb{F} $, where $V$ is a vector space over $\mathbb{F}$ (often taken to be a tangent space). The transformation law then follows.

My current understanding of spinors feels like the first, dissatisfying definition: they're just "things that transform like spinors" - that is, they're elements of a vector space which transform according to a projective representation of $SO(n)$ which is genuinely multi-valued (i.e. it's not just a true representation of $SO(n)$). We could call this the "spinor transformation law". Note that this is something we've put in "by hand": the way a spinor transforms is not a property of some underlying object, but is built into our definition.

My question is: can we define spinors without reference to the way they transform, just as we did for tensors? Is there some object "underlying" the definition of spinors in terms of transformations, just as tensors are "really" multilinear maps?

\begin{align}
\text{Tensor Transformation Law}&\to \text{Tensors as multilinear maps}\\
\text{Spinor Transformation Law}&\to \text{??? }
\end{align}

Answer: 
Yes. Spinors are elements of representation spaces of objects known as a [Clifford algebras](https://en.wikipedia.org/wiki/Clifford_algebra).

A Clifford algebra is basically a vector space turned into an algebra via the product rule

$$
v\cdot w=2g(v,w)\Bbb{1}
$$

where $g$ is some metric on the vector space itself. The most famous Clifford algebra is the Dirac algebra, i.e. the algebra of the Dirac matrices (for which the vector space is $\Bbb{R}^{4}$ and the metric is the Minkowski metric). If instead you use $\Bbb{R}^{3}$ as the base vector space, with the Euclidean metric, you obtain the Pauli algebra.

Once you have a Clifford algebra you can look for its representations (or "modules", as they are known in the literature). The elements of these representations are the [spinors](https://en.wikipedia.org/wiki/Clifford_algebra#Spinors). The spinors corresponding to $\Bbb{R}^{4}$ with the Minkowski metric are the Dirac spinors, whereas those corresponding to $\Bbb{R}^{3}$ with the Euclidean metrics are the spinors of $SO(3)$/$SU(2)$.
Well, you should look at (irreducible) representations of the Lorentz group.
Basically you want all of your ingredients to have correct and consistent transformations under the Lorentz group. 

**The Weyl and Dirac spinors are the most basic objects satisfying that requirement.**

Starting from those you can build vectors as a (multiplicative) combos of two spinors. That's why in old texts you sometimes see spinors referred to as 'half-vectors'. Also, in this context, they use only 'half' of the transformation of a vector, i.e. one-sided vs two-sided.

In that sense its Spinors->Vectors->Tensors.

If you feel fancy, you can also look at things in the context of Geometric Algebra or Spacetime Algebra going back to David Hestenes. Here you can have spinors free of any matrix representation.

Two other references with different perspectives also come to mind: Spinors and space-time (Penrose) and GRAVITATION (Misner Thorne Wheeler)

**The common theme of all approaches is however the special, fundamental transformation properties they have.** You can't get around that.