We calculate the spectral density and occupations of a system of two capacitively coupled quantum dots, each one connected to its own pair of conducting leads, in a regime of parameters in which the total couplings to the leads for each dot Γ_i are different. The system has been used recently to perform pseudospin spectroscopy by controlling independently the voltages of the four leads. For an odd number of electrons in the system, equal coupling to the leads Γ₁ = Γ₂, equal dot levels E₁ = E₂ and sufficiently large interdot repulsion U₁₂ the system lies in the SU(4) symmetric point of spin and pseudospin degeneracy in the Kondo regime. In the more realistic case Γ₁ ≠ Γ₂, pseudospin degeneracy is broken and the symmetry is reduced to SU(2). Nevertheless, we find that the essential features of the SU(4) symmetric case are recovered by appropriately tuning the level difference δ = E₁-E₂. After this tuning, the system behaves as an SU(4) Kondo one at low energies. Our results are relevant for experiments which look for signatures of SU(4) symmetry in the Kondo regime of similar systems.