Electronic – Square wave inverter

You can use a 2nd order low pass filter made from an inductor (in series with the AC) and a capacitor to neutral. You will also need a resistor: -

^{simulate this circuit – Schematic created using CircuitLab}

1 Henry and 10uF form a 2nd order low pass filter that will attenuate the higher order frequencies from the sq wave and make the output signal more sinusoidal in nature. It might work fine or it still might give your TV problems if it is particularly sensitive to harmonics.

The resistor is needed to keep the Q of the circuit low and prevent high ringing voltages.

Q = \$\dfrac{1}{R}\sqrt{\dfrac{L}{C}}\$ = 3.16 using the values above.

Q needs to be about unity so I'd advise to make C bigger and L proportionally smaller but be aware that the value of resistance is in series with the TV and whatever current the TV takes, it flows through the resistor. For instance, 110 watts needs 0.5 amps from 220 V and this means the I^2R losses in the resistor are 25 watts - choose big wirewound resistors.

If it is 'same direction' would it have any relation with primary windings direction?

The two windings are wound in the same direction just like this little ferrite wound component: -

If the centre-tap wasn't there the orange and red would be one continuous winding in the same direction.

The direction that both have in reference to the primary winding is of no-importance for AC power transformers but, in other applications like transformer fed loudspeaker arrays it would make a massive difference at bass frequencies if one transformer had the opposite phase to another.

Also, is there any change required in terms of winding direction, if used in a Full Bridge inverter?

No.