Suppose you have a random sample of uncirculated US double eagle gold coins which you have assayed for gold content. Although manufacturing variations precluded the US Mint from guaranteeing that every double eagle had exactly 0.9675 Troy ounces of gold, the Mint tried to achieve this as an average value. If they succeeded, the distribution of weights in your sample should be centered close to this value. A sample average that is sufficiently far from 0.9675 (compared to the spread of your measured values) would be evidence that these coins are not honest double eagles.
In this example the null hypothesis is that the population of double eagles averages 0.9675 ounces and the alternative is that the average differs from 0.9675. Suppose you were to swap these two statements and instead tried to test whether the population mean differs from 0.9675. You cannot test this hypothesis with data because (literally) any set of values would be consistent with it. (If you always obtained 0.9675 in every assay, your measurement procedure itself would be called into question because the results would be too consistent!) There is an inherent, profound, asymmetry between the two hypotheses because one of them makes a specific quantitative prediction about how the data might be distributed but the other does not.
There's another asymmetry in hypothesis testing. In the same situation, you might be interested in assessing whether the population of US double eagles is underweight. A sample average that is sufficiently low would be good evidence of that. The average would have to be substantially lower than 0.9675, though: how much lower is the "critical value" for the test. In this situation you can switch the null and alternate hypotheses. The new null is that the population of US double eagles is overweight. A sample average that is sufficiently high would be good evidence of that. That average would have to be substantially higher than 0.9675, though: how much higher is the critical value for this reversed test. In each case, the set of possible sample averages is partitioned into two parts: those less than the critical value and those greater than the critical value. Because the two critical values are not the same, the partitions differ, too. For instance, a sufficiently low sample average in the first case lets you conclude that the population mean is underweight, but in the second case it is consistent with the null hypothesis that the population mean is not overweight. Notice the distinction between evidence that is consistent with a hypothesis and evidence that falsifies a hypothesis. That asymmetry is inherent in the logic of hypothesis testing.