Is it an okay idea to swish for two minutes with baking soda, then brush teeth for two minutes afterwards, or the reverse?

No problem with either. Baking Soda is mildly abrasive and is an Acid Neutralizing solution which helps prevent decay. ARM & HAMMER makes toothpaste with Baking Soda as main ingredient. Because baking soda is not acidic, in fact it is acid neutralizing, it will not soften the enamel. Acids like fruit juices makes teeth more likely to abrade away with brushing.

TAKE THE BAKING SODA AND PLACE IT ON YOUR BRUSH AND BRUSH ,OR BRUSH WITH TOOTH PASTE OR A COMBINATION OF BOTH. YOU DON'T NEED TO RINSE YOU HAVE INGREDIENTS TO GET THE JOB DONE. IF YOU EAT A LOT OF ACIDIC FOODS AND FRUITS THAN RINSE WITH BAKING SODA.

Baking soda is a mild abrasive, sometimes used in toothpastes. Swishing with baking soda is generally ineffective. Brushing for 2 mins 2x/day, w or w-out t'paste, will remove 2/3 of harmful bacteria. Flossing or using WaterPik will remove the other 1/3. Baking Soda will not. Baking Soda, mouthwash, oil pulling, are not substitutes for flossing.