# [Python 3], <sup><s>199</s></sup> 197 bytes

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    f=lambda s,n=0:s[0]=='-'and f(s[1:],1-n)or len(s)>1and f({'1i':'i','1j':'j','1k':'k','ii':'-1','jj':'-1','kk':'-1','ij':'k','jk':'i','ki':'j','ji':'-k','kj':'-i','ik':'-j'}[s[:2]]+s[2:],n)or'-'*n+s

[Try it online!][TIO-kp5t4uq8]

[Python 3]: https://docs.python.org/3/
[TIO-kp5t4uq8]: https://tio.run/##bVDBbsMgDL33K3wjrIlU0kmTItEfyXLoEqIatzQqTNoW5dszTOhlmgXWs5/9bJi@w@Xujus66uv59jGcwZdOHxrfHjqtRSXOboCx8K1qulJVTt4fcDWu8PKkNmoWCkUjUJRC2QgsA4qAIkCmKhWRtU9E9ERoc5mlrECYFWxqZI5SI5OYGq1YWt82ddftfVvHrXinuOiL2/s1GB88aChYDVgJeADwKI5TlnKGUg6R2OxWi2TTTYHlGqTsN4FYiJRPNCF35msyfTDDf2OTqzavkrd/0mprkLv@Ynoyj6Ty/lm/vfbMMFLHSI/x40NpAB384FSkh5bwnC2bHUTjp49FkCmYHuhCMYo5LFCdYPYLzHlIa7T23SLk@gs "Python 3 – Try It Online"  

*Quite* verbose!