Cycling clothing care products
The fundamental problem with synthetic garments is that some of the fibres (particularly early nylon and current polyester) have spaces within the fibres that are perfect-sized cocoons for growing the bacteria that give rise to body odour. 'Boil wash' the garment to kill the bacteria and you damage the fabric. Cold-wash to look after the fabric and you don't kill the bacteria. It means you cannot win on a regular machine wash.

As mentioned vinegar (an acid) works to kill the bacteria. The same goes with a domestic alkali such as baking soda. If this pre-wash is too much effort and you just want the bacteria killed then you can get TAED or 'oxy' add-in for your wash - same aisle as the regular washing powders, smaller packs. This works at low temperatures and with my clothes but there is no way in a million years that I would advise an 'Assos' garment (or equivalent) be washed with TAED.

You can also invest in the base-layers that have silver particles embedded in them. These are up-market garments that cost a bit more but pay off in the no-smells department.

Also of use is a dehumidifier, the DeLonghi DNC65 will work wonders in the changing room at work or at home in the winter months to purge moisture from the air where your clothing develops its pong. A lot of life's problems were solved with the DNC65 for me and my invariably damp cycle attire.

The aim of the game is to not have the bacteria grow, it is therefore best to shower before riding to work (not at work) as fresh sweat does not smell - just the bacteria that develop 12 hours later (when you are home). Therefore, clean body and no bacteria gets transferred to the clothing - it stays that bit cleaner.

I don't think anything can be done with cheaper fabrics that get infested with bacteria - a good set of quality items, washed as per the instructions and kept scrupulously dry in-between rides with a bit of 'oxy' added to the wash is the best strategy. You also want to avoid merely disguising smells - the deodorant strategy - as that does not really work.