Why are test hypotheses not symmetric? Suppose,
\begin{align}
H_0\!: p<0  \\
H_1\!: p\geq0
\end{align}
and further that I measure $-100$. Why am I not allowed to say that $H_0$ is true?
If you switch null and alternative hypotheses:
\begin{align}
H_0\!: p\geq0  \\
H_1\!: p<0
\end{align}
and I measure $-100$, I can conclude that $p<0$. 
I do not understand that by just reversing the hypotheses I can draw the conclusion according to what I measured.
The following case is clear to me: 
\begin{align}
H_0\!: p=0  \\
H_1\!: p\neq0
\end{align}
If you measure $0.1$ for example I understand you do not accept $H_0$, because $p$ could be $0.5$ or $-0.03$ etc.  But if you measured $100$, you would reject $H_0$. 
 A: If we can show that the observed data are inconsistent with a given hypothesis, then that is grounds for rejecting that hypothesis.  However, merely showing that the data are consistent with a hypothesis is not reason to assume that such a hypothesis is true.  This kind of thinking leads to contradictions because the same data can be consistent with multiple, mutually exclusive hypotheses.
This is primarily true when testing so-called point null hypotheses where we suppose that a parameter equals some specific value.  Typically when the data are consistent with the null value they will also be consistent with a range of other values close to the null as well.  In your example you would still need to show that the data are not consistent with the alternative in order to conclude that the null is true.
Another circumstance where we might accept even point null hypotheses is when the range of plausible values are practically indistinguishable from the null value.  This is usually due to a large sample size and happens when you have a confidence interval that's very tight around the null value.
