  We derive various properties of symmetric product orbifolds of $T\bar{T}$ and
$J\bar{T}$ - deformed CFTs from a field-theoretical perspective. First, we
generalise the known formula for the torus partition function of a symmetric
orbifold theory in terms of the one of the seed to non-conformal
two-dimensional QFTs; specialising this to seed $T\bar{T}$ and $J\bar{T}$ -
deformed CFTs reproduces previous results in the literature. Second, we show
that the single-trace $T\bar{T}$ and $J\bar{T}$ deformations preserve the
Virasoro and Kac-Moody symmetries of the undeformed symmetric product orbifold
CFT, including their fractional counterparts, as well as the KdV charges.
Finally, we discuss correlation functions in these theories. By extending a
previously-proposed basis of operators for $J\bar{T}$ - deformed CFTs to the
single-trace case, we explicitly compute the correlation functions of both
untwisted and twisted-sector operators and compare them to an appropriate set
of holographic correlators. Our derivations are based mainly on Hilbert space
techniques and completely avoid the use of conformal invariance, which is not
present in these models.
