A hybrid transmitting and reflecting beyond diagonal reconfigurable intelligent surface (BD-RIS) design is proposed. Operating in the same aperture, frequency band and polarization, the proposed BD-RIS features independent beam steering control of its reflected and transmitted waves. In addition it provides a hybrid mode with both reflected and transmitted waves using tunable power splitting between beams. The BD-RIS comprises two phase reconfigurable antenna arrays interconnected by an array of tunable two-port power splitters. The two-port power splitter in each BD-RIS cell is built upon a varactor in parallel with a bias inductor to exert tunable impedance variations on transmission lines. Provided with variable reverse DC voltages, the two-port power splitter can control the power ratio of S11 over S21 from -20 dB to 20 dB, thus allowing tunable power splitting. Each antenna is 2-bit phase reconfigurable with 200 MHz bandwidth at 2.4 GHz so that each cell of BD-RIS can also achieve independent reflection and transmission phase control. To characterize and optimize the electromagnetic response of the proposed BD-RIS design, a Thévenin equivalent model and corresponding analytical method is provided. A BD-RIS with 4 by 4 cells was also prototyped and tested. Experiments show that in reflection and transmission mode, the fabricated BD-RIS can realize beam steering in reflection and transmission space, respectively. It is also verified that when operating in hybrid mode, the BD-RIS enables independent beam steering of the reflected and transmitted waves. This work helps fill the gap between realizing practical hardware design and establishing an accurate physical model for the hybrid transmitting and reflecting BD-RIS, enabling hybrid transmitting and reflecting BD-RIS assisted wireless communications.