Your doctor will have to decide that for you. They are not equivalent medications though. Asmanex is only an inhaled corticosteroid (ICS). Symbicort is a combination medication with an inhaled corticosteroid and a long-acting beta2 agonist (LABA - bronchodilater).
My limited understanding is that COPD requires both types of medications. Sometimes an LABA can be used to treat COPD as a stand alone medication (always in combination for an asthmatic). Sevevent is a stand alone LABA if you want to try that. I doubt that your doctor will take you off the Symbicort.