We recently described a specific type of attractors of two-dimensional discontinuous piecewise linear maps, characterized by two discontinuity lines dividing the phase plane into three partitions, related to economic applications. To our knowledge, this type of attractor, which we call a weird quasiperiodic attractor, has not yet been studied in detail. It has a rather complex geometric structure and other interesting properties that are worth exploring in more depth. To this end, we consider a simpler map that can also possess weird quasiperiodic attractors, namely, a 2D discontinuous piecewise linear map with a single discontinuity line dividing the phase plane into two partitions, where two different homogeneous linear maps are defined. Map depends on four parameters — the traces and determinants of the two Jacobian matrices. In the parameter space of map , we obtain specific regions associated with the existence of weird quasiperiodic attractors; describe some characteristic properties of these attractors; and explain one of the possible mechanisms of their appearance.