"Rejser" is the present tense of "at rejse", "to travel". So there is no clear path to "I get up" from what looks like "I travel me". This is unlike "lide" meaning both "like" and "suffer", where "Jeg kan godt lide noget" could be interpreted as "I can well suffer something".
"at rejse" thus "rejser" have two distinct verbal meanings
https://ordnet.dk/ddo/ordbog?select=rejse,2&query=rejse -> to raise something up in an upright position. "to hoist" or "to rise"
The reflexive verb 'at rejse sig' (to get up) has a different meaning to 'at rejse' (to travel). Some verbs of movement are reflexive in Danish, but 'rejse' isn't one of them. As well as that, 'to travel', in English, is intransitive (usually) and non-reflexive, so 'I travel myself' doesn't have a meaning.