Duals of the spinor representations of $\frak{so}_{2n}$

Here is one explanation, although I am not sure if this is the conceptual explanation you are looking for.

Let $E = \mathbb{R}^n$ with orthonormal basis $\varepsilon_1, \ldots, \varepsilon_l$. You can realize a root system of type $D_n$ as $\Phi = \{ \pm (\varepsilon_i \pm \varepsilon_j) : i \neq j \}$.

The Weyl group $W$ is the group of permutations and sign changes on $\varepsilon_1$, $\ldots$, $\varepsilon_n$ involving only an even number of sign changes. That is, for $\sigma \in W$ you have $\sigma(\varepsilon_i) = c_i \varepsilon_{\pi(i)}$ with $\pi \in Sym_n$, $c_i = \pm 1$ and $c_1c_2 \cdots c_n = 1$.

From this you already see that $-1 \in W$ if and only if $n$ is even.

Also, the weights occurring in the two irreducible spin representations are precisely those of the form $$\lambda_I = \frac{1}{2} (\sum_{i \in I} \varepsilon_i - \sum_{i \not\in I} \varepsilon_i)$$ for a subset $I \subseteq \{1, \ldots, n\}$. Clearly $\lambda_I$ and $\lambda_{J}$ are conjugate under the Weyl group if and only if $|I| = |J| \mod{2}$.

One of the spin representations has highest weight $\lambda = \frac{1}{2}(\sum_{i = 1}^n \varepsilon_i)$ and the other one has highest weight $\mu = \frac{1}{2}(\sum_{i = 1}^{n-1} \varepsilon_i - \varepsilon_n)$. So $\lambda$ and $-\mu$ are conjugate under the Weyl group if and only if $-\mu$ involves an even number of minus signs, equivalently $n$ is odd.