Architected two-dimensional (2D) lattice materials consisting of elastic beams are appealing because of their tunable sign of Poisson's ratio. A common belief is that such materials with positive and negative Poisson's ratios display anticlastic and synclastic curvatures, respectively, when bent in one direction. Here, we show theoretically and demonstrate experimentally that this is not true. For 2D lattices with star-shaped unit cells, we find a transition between anticlastic and synclastic bending curvatures controlled by the beam's cross-sectional aspect ratio even at a fixed Poisson's ratio. The mechanisms lay in the competitive interaction between axial torsion and out-of-plane bending of the beams and can be well captured by a Cosserat continuum model. Our result may provide unprecedented insights to the design of 2D lattice systems for shape-shifting applications.