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A polarized capacitor forms the dielectric between the conducting plates electrochemically. Voltage must be applied in one direction only across the conducting plates to form the dielectric insulating film. The applied voltage across the capacitor must always be in a particular direction, and the capacitor is externally marked to indicate the proper voltage polarity. Reversing the applied voltage can cause the insulating film to rapidly fail, and can cause the capacitor to actually explode from rapid production of gas within the shell of the capacitor.
A non-polarized capacitor uses a dielectric material with uniform insulating properties (such as mylar, mica, glass, etc.), and can be installed in a circuit without regard for the polarity of the applied voltage.
You can use caps back to back for non-critical applications, it is better to use them in conjunction with the diodes, but their characteristics will never be exactly the same as a true non-polarized cap.
At one time these tricks with polar ones were necessary. Now the nonpolar ones are so common that Digikey and other mail order sources sell them for modest prices. In years gone by, these were used in audio power amplifiers which were powered from one polarity of supply.