  As a consequence of the Schwartz kernel Theorem, any linear continuous
operator $\widehat{A}:$
$\mathcal{S}(\mathbb{R}^{n})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n})$
can be written in Weyl form in a unique way, namely it is the Weyl quantization
of a unique symbol $a\in\mathcal{S}^{\prime}(\mathbb{R}^{2n})$. Hence,
dequantization can always be performed, and in a unique way. Despite the
importance of this topic in Quantum Mechanics and Time-frequency Analysis, the
same issue for the Born-Jordan quantization seems simply unexplored, except for
the case of polynomial symbols, which we also review in detail. In this paper
we show that any operator $\widehat{A}$ as above can be written in Born-Jordan
form, although the representation is never unique if one allows general
temperate distributions as symbols. Then we consider the same problem when the
space of temperate distributions is replaced by the space of smooth slowly
increasing functions which extend to entire function in $\mathbb{C}^{2n}$, with
a growth at most exponential in the imaginary directions. We prove again the
validity of such a representation, and we determine a sharp threshold for the
exponential growth under which the representation is unique. We employ
techniques from the theory of division of distributions.
