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A new high order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions. The staggered DG scheme defines the discrete pressure on the primal triangular mesh, while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh. In this paper, a new pair of equal-order-interpolation velocity-pressure finite elements is proposed. On the primary triangular mesh (the pressure elements) the basis functions are piecewise polynomials of degree $N$ and are allowed to jump on the boundaries of each triangle. On the dual mesh instead (the velocity elements), the basis functions consist in the union of piecewise polynomials of degree $N$ on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries, while they are continuous inside. In other words, the basis functions on the dual mesh are built by continuous finite elements on the subtriangles. This choice allows the construction of an efficient, quadrature-free and memory saving algorithm. In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations, arbitrary high order of accuracy in time is achieved through the use of time-dependent test and basis functions, in combination with simple and efficient Picard iterations. Several numerical tests on classical benchmarks confirm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes, not only from a computer memory point of view, but also concerning the computational time.