Prompted by the lack of a unique choice of pressure (P) and density (ρ) fields for a compressible free vortex and by the observed dependence of turbulence dynamics on initial P and ρ in compressible simulations, we address the effects of initial conditions on the evolution of a single vortex, on the prototypical phenomenon of vortex reconnection, and on two-dimensional turbulence. Two previous choices of initial conditions used for numerical simulations of compressible turbulence have been: (i) both P and ρ uniform (constant initial conditions, CIC), and (ii) uniform ρ with P determined from the Poisson equation (constant density initial conditions, CDIC). We find these initial conditions to be inappropriate for compressible vorticity dynamics studies. Specifically, in compressible reconnection, the effects of baroclinic vorticity generation and shocklet formation cancel each other during early evolution for CDIC, thus leading to almost incompressible behavior. Although CIC captures compressibility effects, it incorrectly changes the initial vorticity distribution by introducing strong acoustic transients, thereby significantly altering the evolving dynamics. Here, a new initial condition, called polytropic initial condition (PIC), is proposed, for which the Poisson equation is solved for initially polytropically related P and ρ fields. PIC provides P and ρ distributions within vortices which are consistent with those observed in shock-wedge interaction experiment and also leads to compressible solutions with no acoustic transients. At low Mach number (M), we show that the effects of all these three initial conditions can be predicted by low-M asymptotic theories of the Navier-Stokes equations. At high M, it is shown here that inappropriate initial conditions may alter the evolutionary dynamics and, hence, lead to wrong conclusions regarding compressibility effects. We argue that PIC is a more appropriate choice.