What is the dual of an semidefinitely representable (SDR) cone? The Question
Let $V\simeq \mathbb{R}^r$ be an $r$-dimensional vector space with the usual Euclidean inner product. 
Let $\mathcal K\subset V$ be a cone defined as
$$
    \mathcal K=\Big\{x\in V\ \Big|\ \exists y\in\mathbb{R}^s~\mathrm{such~that}~\sum_{i=1}^r x_i X_i+\sum_{j=1}^s y_j Y_j\geq0\Big\},
$$
where $\{X_i\}_{1\leq i\leq r},\{Y_j\}_{1\leq j\leq s}\in B(H)^{sa}$ are self-adjoint operators acting on some finite-dimensional Hilbert space $H$. These cones are called semidefinite representable (SDR). They are projections of spectrahedra and have all the nice properties of the latter, plus some others. Some good reference on these beasts would be great.
The dual of $\mathcal K$ is defined as $\mathcal K^*=\{x'\in V\ |\ x\cdot x'\geq0~\forall x\in \mathcal K\}$.

Question: Is $\mathcal K^*$ an SDR cone? What are the corresponding operators?

In particular I am interested in an answer presented in the same form, i.e. some set of self-adjoint operators defining $\mathcal K^*$. However, but I'm not sure it's even possible.
Of course a general solution would be great, but I can settle by making a few assumptions.
Assumption 0: Without loss of generality, $\{Y_j\}$ can be assumed to be linearly independent.
Assumption 1: The cone $\mathcal K$ is pointed or salient; $\mathcal K\cap(-\mathcal K)=\{0\}$.
Assumption 2: The cone $\mathcal K$ is generating; $V=\mathcal K-\mathcal K$.

Some partial insights
Assumption 1 already tells us some things about $\{X_i\}$ and $\{Y_j\}$. A few immediate consequences are


*

*Fact 1: $\{X_i\}$ are linearly independent. If they weren't $\mathcal K$ would contain some entire subspace of $V$, thus wouldn't be pointed.

*Fact 2: $\mathrm{span}\{Y_i\}$ does not contain an order unit for $\mathrm{span}\{X_i\}$ or any subspace thereof. Same reason as above.



  
*
  
*Fact 3: $\{X_i,Y_j\}$ are linearly independent. 
  

Proof: Suppose they are not. Take 
$$\sum_i \alpha_i X_i+\sum_j \beta_j Y_j=0$$ 
with some nonzero coefficients, so that $\sum_i \alpha_i X_i=-\sum_j \beta_j Y_j$ is nonzero because $\{X_i\}$ and $\{Y_j\}$ are linearly independent. Then $\alpha\neq0$. For any $\lambda\in\mathbb{R}$, $\lambda\alpha\in\mathcal K$. Thus $\mathcal K$ is not pointed. $\blacksquare$
From Fact 3 we can complete the set of operators to $\{X_i,Y_j,Z_k\}_{(1\leq i\leq r, 1\leq j\leq s, 1\leq k\leq t)}$ to form a basis of $B(H)^{sa}$. In addition, define the conjugate basis with respect to the Hilbert-Schmidt inner product
\begin{align}
\begin{array}{ccc}
\mathrm{tr}[X_i \tilde X_{i'}]=\delta_{ii'}, &\mathrm{tr}[X_i \tilde Y_{j'}]=0&\mathrm{tr}[X_i \tilde Z_{k'}]=0\\\
\mathrm{tr}[Y_j \tilde X_{i'}]=0, &\mathrm{tr}[Y_j \tilde Y_{j'}]=\delta_{jj'}&\mathrm{tr}[Y_j \tilde Z_{k'}]=0\\\
\mathrm{tr}[Z_k \tilde X_{i'}]=0, &\mathrm{tr}[Z_k \tilde Y_{j'}]=0&\mathrm{tr}[Z_k \tilde Z_{k'}]=\delta_{kk'}\\\
\end{array}
\end{align}

Partial answer: With the conjugate basis one can define 
  $$
    \mathcal C=\Big\{a\in V\ \Big|\ \exists c\in\mathbb{R}^t~\mathrm{such~that}~\sum_{i=1}^r a_i \tilde X_i+\sum_{k=1}^t c_k \tilde Z_k\geq0\Big\}.
$$
  and show that $\mathcal C\subseteq\mathcal K^*$. 

Proof: Let $a\in\mathcal C$. Then there is  $c\in\mathbb{R}^t$ such that
$$
     \mathcal A=\sum_{i}a_{i} \tilde X_{i}+\sum_{k}c_{k} \tilde Z_k\geq0.
$$
For any $x\in\mathcal K$, there is $y\in\mathbb{R}^s$ such that
$$
     \mathcal X=\sum_{i}x_i X_i+\sum_{j}y_j Y_j\geq0   
$$
thus the inner product $x\cdot a=\mathrm{tr}[\mathcal X\mathcal A]\geq0$. Therefore, 
$$
  a\in\mathcal C~~\Rightarrow~~ x\cdot a\geq0~\forall x\in\mathcal K~~\Rightarrow~~ a\in\mathcal K^*.
$$

Alternative Question: Under what conditions it is true that $\mathcal K^*=\mathcal C$ ?

[Edit] Partial answer: A sufficient condition for equality is that $\mathrm{span}\{X_i,Y_j\}$ intersects the interior of the positive semidefinite cone, as Noah explains in his answer. Whether Assumption 2 guarantees this is an open question (for me). Interestingly, for spectrahedra, this is always the case.
 A: Edited in response to Alex Monras's correction in the comments:
The cone $\mathcal{K}^*$ is always SDR: this is just the conic / homogeneous version of Theorem 5.57 in the new book "Semidefinite Optimization and Convex Algebraic Geometry" by Blekherman, Parrilo, and Thomas.  This book is also a good source for more background about this field.
As for the proof, it consists of writing the semidefinite program: minimize $Y\bullet X$ subject to $X\in\mathcal{K}$.  This is always feasible (take $X=0$) and so has optimum either $0$ or $-\infty$.  Take the usual dual semidefinite program.  If strong duality holds (e.g. there is an $(x,y)$ pair making the constraint strictly positive definite), the dual has zero objective and the set of $Y$ for which it is feasible defines exactly the semidefinite representation of $\mathcal{K}^*$.
To show $\mathcal{K} ^ * $ is SDR without imposing a constraint qualification like the Slater condition, you need an extended dual SDP construction like the one proposed by Ramana.  In either case, working through the details of the proof will (at least in principle) give the precise form of the representation for $\mathcal{K}^*$.
A: It's a long comment, not an answer. 
Already the well-known SDR cone $\Sigma_{n,d}$ of sums of squares of homogeneous $n$-variate degreed $d$ polynomials has an interesting dual (also SDR), described e.g. in B.Reznick's AMS Memoir, Theorem 3.16. At least this gives interesting examples to play with.
