Investigations of structural and magnetic properties of polycrystalline hole doped double perovskite La_1.5Ca_0.5CoMnO₆ has clearly revealed the existence of structural antisite-disorder (either, Co–O–Co or Mn–O–Mn) in the system. The ordering of Co²⁺ and Mn⁴⁺ gives rise to a ferromagnetic transition around 157 K. A spin-canted antiferromagnetic transition is found in this material at TCAFM ~9 K. The effect of antisite-disorder in the double perovskite structure is most likely the prime reason for antiferromagnetic interaction. The temperature dependent inverse susceptibility exhibits Curie-Weiss like behaviour and it yields an effective paramagnetic moment of 6.49 μB. At very low temperature (T < TCAFM), it shows a strong hysteresis loop shift with large exchange bias (EB) field of HEB ~5.5 kOe and can be tuned by the cooling field. The presence of zerofield cooled spontaneous EB effect (P-type and N-type) is confirmed to be not an experimental artefact - an inherent property of this double perovskite material. A phenomenological model has been proposed to explain the exchange coupling between the ferromagnetic and canted-antiferromagnetic interfaces of antisite disordered La_1.5Ca_0.5CoMnO₆ mainly on the basis of uncompensated interface spins.