Let me try to answer this question. I apologise if my notation is slightly different, since I will work in some more generality, since the equivariance properties of the creation and annihilation operators are actually more transparent, I believe, relative to the the general linear group instead of the orthogonal group.

Let $E$ be an $n$-dimensional real vector space and let $E^*$ denote its dual. Then on $H = E \oplus E^* \oplus \mathbb{R}K$ one defines a Lie algebra by the following relations: $$ [x,y]= 0 = [\alpha,\beta] \qquad [x,\alpha] = \alpha(x) K = - [\alpha,x] \qquad [K,*]=0$$ for all $x,y \in E$ and $\alpha,\beta \in E^*$. This is called the Heisenberg Lie algebra of $E$, denoted $\mathfrak{h}$.

The automorphism group of $\mathfrak{h}$ is the group $\operatorname{O}(E\oplus E^*)$ of linear transformations of $E\oplus E^*$ which preserve the split-signature inner product defined by the dual pairing: $$\langle (x,\alpha), (y,\beta) \rangle = \alpha(y) + \beta(x).$$

Let $\mathfrak{a} < \mathfrak{h}$ denote the abelian subalgebra with underlying vector space $E \oplus \mathbb{R}K$. One can induce a $\mathfrak{h}$-module from an irreducible (one-dimensional) $\mathfrak{a}$-module as follows. Let $W_k$ denote the one-dimensional vector space on which $E$ acts trivially and $K$ acts by multiplication with a constant $k$. Then letting $U$ be the universal enveloping algebra functor, we have that $$ V_k = U\mathfrak{h} \otimes_{U\mathfrak{a}} W_k$$ is an $\mathfrak{h}$-module. The Poincaré-Birkhoff-Witt theorem implies that $V_k$ is isomorphic as a vector space to the symmetric algebra of $E^*$, which we may (as we are over $\mathbb{R}$) identify with polynomial functions on $E$.

The subgroup of $\operatorname{O}(E\oplus E^*)$ which acts on $V_k$ is the general linear group $\operatorname{GL}(E)$ and hence $V_k$ becomes a $\operatorname{GL}(E)$-module. In fact, $V_k$ is graded (by the grading in the symmetric algebra of $E^*$ or equivalently the degree of the polynomial): $$V_k = \bigoplus_{p\geq 0} V_k^{(p)}$$ and each $V_k^{(p)}$ is a finite-dimensional $\operatorname{GL}(E)$-module isomorphic to $\operatorname{Sym}^p E^*$.

Every vector $x \in E$ defines an annhilation operator: $A(x): V_k^{(p)} \to V_k^{(p-1)}$ via the contraction map $$E \otimes \operatorname{Sym}^p E^* \to \operatorname{Sym}^{p-1} E^*$$ whereas every $\alpha \in E^*$ defines a creation operator: $C(\alpha): V_k^{(p)} \to V_k^{(p+1)}$ by the natural symmetrization map $$E^* \otimes \operatorname{Sym}^p E^* \to \operatorname{Sym}^{p+1} E^*.$$

Both of these maps are $\operatorname{GL}(E)$-equivariant, and this is perhaps the most invariant statement I can think of concerning the creation and annihilation operators.

Now, of particular interest are the unitary representations of $\mathfrak{h}$ and this is in any case the original setting of this question. I will not do this in detail, but it is not hard. Just introduce a positive-definite inner product on $E$, hence on $\operatorname{Sym}^pE^*$ for every $p$, whence ultimately on $V_k$ after a choice of inner product on $W_k$. It is usual to make $W_k$, and hence $V_k$, into a complex representation, in which case the inner product extends to a hermitian inner product on $V_k$. Finally, it is then usual to consider the Hilbert space completion of $V_k$. Then it is the orthogonal group $\operatorname{O}(E)$ preserving the inner product on $E$ which acts on $V_k$ preserving the inner product. The inner product on $E$ also defines musical isomorphisms between $E$ and $E^*$, whence we can attach to every $x \in E$ not just an annihilation operator, but also a creation operator. The equivariance properties I mentioned above now restrict to equivariance under $\operatorname{O}(E)$.

Let me try to answer this question. I apologise if my notation is slightly different, since I will work in some more generality, since the equivariance properties of the creation and annihilation operators are actually more transparent, I believe, relative to the the general linear group instead of the orthogonal group. Also the fact that this is the harmonic oscillator is a red herring. In the case of the harmonic oscillator, we have to introduce more structure, reducing the group of symmetries. It is also in this case, where the grading to be discussed below coincides (up to a choice of scale) with the energy of the system.

Let $E$ be an $n$-dimensional real vector space and let $E^*$ denote its dual. Then on $H = E \oplus E^* \oplus \mathbb{R}K$ one defines a Lie algebra by the following relations: $$ [x,y]= 0 = [\alpha,\beta] \qquad [x,\alpha] = \alpha(x) K = - [\alpha,x] \qquad [K,*]=0$$ for all $x,y \in E$ and $\alpha,\beta \in E^*$. This is called the Heisenberg Lie algebra of $E$, denoted $\mathfrak{h}$.

The automorphism group of $\mathfrak{h}$ is the group $\operatorname{Sp}(E\oplus E^*)$ of linear transformations of $E\oplus E^*$ which preserve the symplectic inner product defined by the dual pairing: $$\omega\left( (x,\alpha), (y,\beta) \right) = -\alpha(y) + \beta(x).$$

Let $\mathfrak{a} < \mathfrak{h}$ denote the abelian subalgebra with underlying vector space $E \oplus \mathbb{R}K$. One can induce a $\mathfrak{h}$-module from an irreducible (one-dimensional) $\mathfrak{a}$-module as follows. Let $W_k$ denote the one-dimensional vector space on which $E$ acts trivially and $K$ acts by multiplication with a constant $k$. Then letting $U$ be the universal enveloping algebra functor, we have that $$ V_k = U\mathfrak{h} \otimes_{U\mathfrak{a}} W_k$$ is an $\mathfrak{h}$-module. The Poincaré-Birkhoff-Witt theorem implies that $V_k$ is isomorphic as a vector space to the symmetric algebra of $E^*$, which we may (as we are over $\mathbb{R}$) identify with polynomial functions on $E$.

The subgroup of $\operatorname{Sp}(E\oplus E^*)$ which acts on $V_k$ is the general linear group $\operatorname{GL}(E)$ and hence $V_k$ becomes a $\operatorname{GL}(E)$-module. In fact, $V_k$ is graded (by the grading in the symmetric algebra of $E^*$ or equivalently the degree of the polynomial): $$V_k = \bigoplus_{p\geq 0} V_k^{(p)}$$ and each $V_k^{(p)}$ is a finite-dimensional $\operatorname{GL}(E)$-module isomorphic to $\operatorname{Sym}^p E^*$.

Every vector $x \in E$ defines an annhilation operator: $A(x): V_k^{(p)} \to V_k^{(p-1)}$ via the contraction map $$E \otimes \operatorname{Sym}^p E^* \to \operatorname{Sym}^{p-1} E^*$$ whereas every $\alpha \in E^*$ defines a creation operator: $C(\alpha): V_k^{(p)} \to V_k^{(p+1)}$ by the natural symmetrization map $$E^* \otimes \operatorname{Sym}^p E^* \to \operatorname{Sym}^{p+1} E^*.$$

Both of these maps are $\operatorname{GL}(E)$-equivariant, and this is perhaps the most invariant statement I can think of concerning the creation and annihilation operators.