Let $$X: \quad Q_1(x)=Q_2(x) = 0 \quad \subset \mathbb{P}^{2n+1},$$ be a smooth complete intersection of two quadrics of odd dimension over a field $k$, not of characteristic $2$. Let $J(X)$ denote the intermediate Jacobian of $X$. When $k$ is algebraically closed, I know that $J(X)$ is isomorphic to the Jacobian of the hyperelliptic curve $$y^2 = \mathrm{det}(xQ_1 + Q_2). \quad (*)$$

Is there a similar description when $k$ is non-algebraically closed?

The reason I ask is that when $k$ is non-algebraically closed, it seems quite possible that we may need to take a quadratic twist of $(*)$. However the only "canonical" twists are either $(*)$ itself or the multiplication by $-1$. So my guess is that $J(X)$ should be isomorphic to the Jacobian of

$$\pm y^2 = \mathrm{det}(xQ_1 + Q_2),$$

however I don't know which sign to take, and it seems quite possible that the choice of sign could even depend on $n$.

Edit: As pointed out by Sasha, I should have explained which definition of the intermediate Jacobian I'm using. Firstly, in

Deligne - Les intersections complètes de niveau de Hodge un.

Deligne showed how to construct the intermediate Jacobian for complete intersections of Hodge level $1$ over fields of characteristic $0$, using a clever trick coming from certain universal properties (in particular, it is not explicit).

In the special case of complete intersections of two quadrics $X$ as above, I believe that one can define $J(X)$ to be the albanese variety of the Fano variety of $(n-1)$-planes inside $X$. This construction should work over fields of characteristic not equal to $2$, and moreover recover Deligne's construction when $\mathrm{char}(k)=0$, though unfortunately I know neither a proof nor a reference for this fact.

I would be happy with an answer which uses either definition, restricting to characteristic $0$ if necessary.