We consider the self-dual Chern-Simons-Schrödinger equation (CSS), also known as a gauged nonlinear Schrödinger equation (NLS). CSS is L 2 L^{2} -critical, admits solitons, and has the pseudoconformal symmetry. These features are similar to the L 2 L^{2} -critical NLS. In this work, we consider pseudoconformal blow-up solutions under m m -equivariance, m ≥ 1 m\geq 1 . Our result is threefold. Firstly, we construct a pseudoconformal blow-up solution u u with given asymptotic profile z ∗ z^{\ast } : \[ [ u ( t , r ) − 1 | t | Q ( r | t | ) e − i r 2 4 | t | ] e i m θ → z ∗ in H 1 \Big [u(t,r)-\frac {1}{|t|}Q\Big (\frac {r}{|t|}\Big )e^{-i\frac {r^{2}}{4|t|}}\Big ]e^{im\theta }\to z^{\ast }\qquad \text {in }H^{1} \] as t → 0 − t\to 0^{-} , where Q ( r ) e i m θ Q(r)e^{im\theta } is a static solution. Secondly, we show that such blow-up solutions are unique in a suitable class. Lastly, yet most importantly, we exhibit an instability mechanism of u u . We construct a continuous family of solutions u ( η ) u^{(\eta )} , 0 ≤ η ≪ 1 0\leq \eta \ll 1 , such that u ( 0 ) = u u^{(0)}=u and for η > 0 \eta >0 , u ( η ) u^{(\eta )} is a global scattering solution. Moreover, we exhibit a rotational instability as η → 0 + \eta \to 0^{+} : u ( η ) u^{(\eta )} takes an abrupt spatial rotation by the angle \[ ( m + 1 m ) π \Big (\frac {m+1}{m}\Big )\pi \] on the time interval | t | ≲ η |t|\lesssim \eta . We are inspired by works in the L 2 L^{2} -critical NLS. In the seminal work of Bourgain and Wang (1997), they constructed such pseudoconformal blow-up solutions. Merle, Raphaël, and Szeftel (2013) showed an instability of Bourgain-Wang solutions. Although CSS shares many features with NLS, there are essential differences and obstacles over NLS. Firstly, the soliton profile to CSS shows a slow polynomial decay r − ( m + 2 ) r^{-(m+2)} . This causes many technical issues for small m m . Secondly, due to the nonlocal nonlinearities, there are strong long-range interactions even between functions in far different scales. This leads to a nontrivial correction of our blow-up ansatz. Lastly, the instability mechanism of CSS is completely different from that of NLS. Here, the phase rotation is the main source of the instability. On the other hand, the self-dual structure of CSS is our sponsor to overcome these obstacles. We exploited the self-duality in many places such as the linearization, spectral properties, and construction of modified profiles.