What is a symmetrical differential amplifier?
Symmetrical differential amplifier is just a differential amplifier but with the following characteristics:

1.The amplifier (MOS or BJT)should has matched transistors(Q1 and Q2 should have the same characteristics e.g W/L ratio, μn*Cox for MOS symmetrical diff amplifier or same beta, Vτ, Is for BJT symmetrical diff amplifier)

2.The amplitude and the frequency of the input signals should be the same for Q1,Q2

3.The load in the two outputs (RD in the figure below or active load in other cases) should have the same value.In the case of resistor load the resistor should have the same value or in the case of active load the transistors should have the same characteristics.If this doesn't occur,we create in the differential amplifier common mode gain,even if we take the 
out differentially, which mean that the noise(noise is the common mode signal in the input) will be amplified!!!

The last figure shows the asymmetry in the resistors(RD, RD+ ΔRD), which creates an assymetric differential amplifier.