While I am not well versed in Latent Profile Analysis, I am familiar with Rasch. In response to bfoste 01's answer, assuming latent class analysis is described correctly, I feel Rasch does provide similar information as LPA, because in the dimensionaality output table, there is information provided about the presence of other constructs (other than that being measured). Rasch is a 1-parameter IRT method, where the data must fit the parameter of the model (which is determined by what the developer sets within the code: what responses are considered indicative of the construct being measured and to what degree it is representative of the construct- from less to more); while LPA, if I understand correctly, is an IRT method that allows the data to inform the model, meaning the parameters of the model can be "massaged" to fit the responses according how the respondents reacted (thus the model is made to fit the respondents). Imagine a ruler: would you rather the ruler keep "inches" the same for everyone (1-parameter: height) or would you like the ruler to respond to the people in your sample (so that if your sample was from a taller group, "inches" would be further apart than if you sampled another group of slightly smaller people). The difficulty with Rasch is that you have to be VERY knowledgeable about what you are trying to measure (which is difficult, given the latent quality of the constructs trying to be measured) AND be willing to go through multiple iterations. Most likely,you will need to go back and rewrite items to better reflect the construct. Item writing is very difficult and to do it well (so it creates a scale) means using respondents authentic language (which means focus groups, cognitive interviewing, and maybe discourse analysis). So it takes a mixed-method approach- which most ppl won't take the time/effort to do.
More information on Rasch can be found at www.rasch.org/rasch.htm for good beginner information. Also,a good primer book on Rasch is by Fox and Bond. The software for using Rasch is Winsteps (the easiest one for scale creation using 1-P...if you're looking to create a measure for a known multi-dimensional construct, there is Facets software.
If LPA is different than that described, I would appreciate feedback on that as well.