  We study a very special class of $T\bar{J}$ deformations of conformal field
theories in two dimensions. While the deformations break the Lorentz symmetry,
they preserve the twisted Lorentz symmetry. The resulting theory has
right-moving Virasoro as well as left-moving translation and left-moving
(chiral) scale symmetry without left-moving special conformal symmetry (nor
left-moving Virasoro symmetry). As in the original $T\bar{J}$ deformations,
they may be regarded as an operator dependent non-local change of coordinates.
We show concrete examples based on worldsheet string theory and discuss how the
non-unitary nature enables us to circumvent various no-go theorems.
