Let us consider real vector bundles, and denote by $V\_k$ the tautological bundle $V\_k\to BO(k)$. From
$$
Thom(V\oplus 1\_{\mathbb{R}}\to X)=\Sigma Thom(V\to X)
$$
and from $j^\*V\_{k+1}=V\_k\oplus1\_{\mathbb{R}}$, where $j\colon BO(k) \to BO(k+1)$ is the canonical embedding, one sees that writing $MO\_k=\Sigma^{-k}Thom(V\_k\to BO(k))$ one gets a sequence of morphisms $MO\_k\to MO\_{k+1}$ and one can define 
$$
MO=\lim\_{\to} MO\_k
$$
This way (and taking infinite suspension) one defines the Thom spectrum $MO$, whcih can therefore informally be thought as the infinite desuspension of the Thom spectrum of the tautological infinite rank vector bundle over $BO(\infty)$. Direct sum of vector bundles and the fact that $j\_{m,n}^\*V\_{m+n}=V\_m\oplus V\_n$, where $j\_{m,n}\colon BO(m)\times BO(n)\to BO(m+n)$ is the canonical embedding makes $MO$ a ring spectrum. 


Apparently, one should be able to do the very same construction by "doubling" the $V\_k$'s, i.e., by considering vector bundles
$$
W\_k=V\_k\oplus V\_k
$$
noticing that $j^\*W\_{k+1}=W\_k\oplus 1\_{\mathbb{R}^2}$ and so $Thom(j^\*W\_{k+1}\to BO(k)) = \Sigma^2 Thom(W\_{k}\to BO(k))$, and then considering the pointed spaces $M\_2O\_k=\Sigma^{-2k}Thom(W\_k\to BO(k))$ and then the colimit 
$$
M\_2O=\lim\_{\to} M\_2O\_k
$$
This again should be a ring spectrum by the same reason as for $MO$.


As, for any real vector space $V$, its double $V\oplus V$ carries a canonical comple structure $J\colon V\oplus V \to V\oplus V$, and since the defining representation $\mathbb{C}^n$ of $U(n)$ restricted to $O(n)$ splits as the sum of two copies of the defining representation $\mathbb{R}^n$ of $O(n)$, one should get a morphism of ring spectra $M\_2O \to MU$. 


By looking at how the $\hat{A}$ polynomial is derived from the Todd polynomial, one would suspect that the analogous morphism
$$
M\_2Spin \to MU
$$
and a compatibility between this, the standard complex orientation of $KU$, the Atiyah-Bott-Shapiro orientation of $KO$ and complexification of vector bundles is what secretely lies behind the Atiyah-Singer formula for the index of a twisted spin complex, making Atiyah-Singer formula a version of the general kind of Hirzebruch-Riemann-Roch formulas one has when dealing with morphisms of generalized cohomology theories and pushforwards (see, e.g., [Panin-Smirnov](https://pdfs.semanticscholar.org/ca93/96835d2ca19d905bcbbe57f74e8aca234900.pdf)). 


However, I've not been able so far to locate something resembling $M\_2O$ or $M\_2Spin$ in the literature I have searched in, so I could possibly be on a false track here.


Is the construction sketched above possibly correct? What is a reference to it? has it a more canonical name than $M\_2O$? is it really related to the construction and relevance of $\hat{A}$ along the lines sketched above?



A Thom spectrum from "doubled" tautological bundles?
I think that both of your examples, $M\_2Spin$ and $M\_2O$, arise naturally in the context of Thom spectra induced by $(B,f)$-structures. Given a $(B,f)$-structure $\mathcal{B}= \{f\_n: B\_n \to BO(n)\}$, the associated Thom spectrum $M\mathcal{B}$ is defined componentwise as:
$$
M\mathcal{B}\_k = Thom(f\_k^\*V\_k\to B\_k),
$$
where the maps $\Sigma M\mathcal{B}\_k \to M\mathcal{B}\_{k+1}$ are given by looking at the pullback square
$\require{AMScd}$
$$
\begin{CD}
\mathbb{R}\oplus f\_k^\*V\_k @>>> f\_{k+1}^\*V\_{k+1}\\
@VVV @VVV \\
B\_k @>>> B\_{k+1}.
\end{CD}
$$
One can 'double' this construction replacing the maps $f\_k$ with $\tilde{f\_k}$ defined to be the composition:
$$
\tilde{B\_{2k}}:=B\_k\overset{f\_k}{\to} BO(k) \overset{\Delta}{\to} BO(k) \times BO(k) \overset{j\_{k,k}}{\to} BO(2k)
$$
and get a $S^2$-$(B,f)$-structure, a $(B,f)$-structure indexed only on even natural numbers, denoted by $2\mathcal{B}$. By definition, the Thom spectrum $M\_2\mathcal{B}$ associated to this new $S^2$-$(B,f)$-structure, is


$$
(M\_2\mathcal{B})\_{2k} = Thom(\tilde{f\_k^\*}V\_{2k}\to \tilde{B\_{2k}}) = Thom(f\_k^\*V\_k\oplus f\_k^\*V\_k \to B\_k)
$$
$$
(M\_2\mathcal{B})\_{2k+1} = \Sigma(M\_2\mathcal{B})\_{2k}.
$$
In your case, $M\_2Spin$ and $M\_2O$, are (as sequential spectra) the Thom spectra associated to the 'doubled' $(B,f)$-structures that classically define $MSpin$ and $MO$, i.e. the $(B,f)$-structures respectively given by the maps $BSpin(k)\to BO(k)$ and ${{\rm id}}\_{BO(k)}$.


All the details and references can be found in the nlab pages [Thom spectrum](https://ncatlab.org/nlab/show/Thom+spectrum) and [G-structure](https://ncatlab.org/nlab/show/G-structure#InTermsOfBfStructures).


